The geodesic flow of a nonpositively curved graph manifold

Christopher B. Croke^{∗} Bruce Kleiner^{†}

November 21, 1999

Abstract

We consider discrete cocompact isometric actions G ↷^{ρ} X where X is a locally compact Hadamard space^{1}, and G belongs to a class of groups (“admissible groups”) which includes fundamental groups of 3-dimensional graph manifolds. We identify invariants (“geometric data”) of the action ρ which determine, and are determined by, the equivariant homeomorphism type of the action G ↷^{∂_{∞}ρ} ∂_{∞}X of G on the ideal boundary of X. Moreover, if G ↷^{ρ_{i}} X_{i} are two actions with the same geometric data and Φ : X_{1} → X_{2} is a Gequivariant quasi-isometry, then for every geodesic ray γ_{1} : [0, ∞) → X_{1}, there is a geodesic ray γ_{2} : [0, ∞) → X_{2} (unique up to equivalence) so that lim_{t→∞} \frac{1}{t}d_{X_{2}}(Φ ∘ γ_{1}(t), γ_{2}([0, ∞))) = 0. This work was inspired by (and answers) a question of Gromov in [Gro93, p. 136].

1. Introduction

As a consequence of the Morse lemma on quasi-geodesics, geodesic flows are especially simple and well understood in the Gromov hyperbolic case :

a. If ϕ : M_{1} → M_{2} is a homotopy equivalence between closed negatively curved manifolds, then there is an orbit equivalence ϕ̂: SM_{1} → SM_{2} between the unit sphere bundles, which covers ϕ up to homotopy [Gro76].

b. If G is a hyperbolic group, G ↷^{ρ_{i}} X_{i} is a discrete, cocompact, isometric action on a Hadamard space X_{i} for i = 1, 2, and Φ : X_{1} → X_{2} is a G-equivariant quasiisometry, then Φ maps each geodesic γ_{1} ⊂ X_{1} to a subset at uniformly bounded Hausdorff distance from a geodesic γ_{2} ⊂ X_{2}. Moreover, Φ induces an equivariant homeomorphism ∂_{∞}Φ : ∂_{∞}X_{1} → ∂_{∞}X_{2} between ideal boundaries, [Gro87].

∗Supported by NSF grants DMS-95-05175 DMS-96-26-232 and DMS 99-71749.

†Supported by a Sloan Foundation Fellowship, and NSF grants DMS-95-05175, DMS-96-26911, DMS-9022140.

1Following [Bal95] we will refer to CAT (0) spaces (complete, simply connected length spaces with nonpositive curvature in the sense of Alexandrov) as Hadamard spaces.

c. When G ↷^{ρ} X is a discrete, cocompact action of a hyperbolic group on a Hadamard space X, then the induced action G ↷^{∂_{∞}ρ} ∂_{∞}X of G on the boundary of X is a finitely presented dynamical system, [Gro87, CDP90].

Naturally one may ask if properties b and c hold without the assumption of Gromov hyperbolicity. It turns out that they do not: one can readily produce examples of pairs of discrete, cocompact, isometric actions G ↷ X_{1}, G ↷ X_{2} where G-equivariant quasi-isometries X_{1} → X_{2} do not induce boundary homeomorphisms^{2} (this was observed independently by Ruane [Rua96]). In [Gro93] Gromov asked whether two actions G ↷ X_{i} induce G-equivariantly homeomorphic boundary actions G ↷ ∂_{∞}X_{i}. The answer to this is also no: S. Buyalo [Buy98] and the authors independently found pairs of actions which induce inequivalent boundary actions^{3}. Finally, we remark that the boundary action G ↷ ∂_{∞}X is finitely presented if and only if G is hyperbolic^{4}.

In this paper we examine actions G ↷ X where G belongs to a class of groups which generalize fundamental groups of 3-dimensional graph manifolds. We develop a kind of “coding” for geodesic rays in X, which allows us to understand the boundary action G ↷ ∂_{∞}X and the Tits metric on ∂_{∞}X. Before stating our main result in complete generality, we first formulate it for nonpositively curved 3-dimensional graph manifolds.

By the theorem of [Sch86], when M is a 3-dimensional graph manifold with a nonpositively curved Riemannian metric, then M has the following structure. There is a collection M_{1}, …, M_{k} of compact nonpositively curved 3-manifolds with nonempty totally geodesic boundary (the geometric Seifert components of M), and Seifert fibrations M_{i} →^{p_{i}} N_{i} where the metric on M_{i} has local product structure compatible with the fibration p_{i}, and the N_{i} are nonpositively curved orbifolds; M is obtained from the disjoint union ⨿_{i} M_{i} by gluing boundary components isometrically in pairs via gluing isometries which are incompatible with the boundary fiberings. In what follows we will only consider graph manifolds whose Seifert fibered components have orientable fiber. Note that for each 1 ≤ i ≤ k, the universal cover of M_{i} is isometric to a Riemannian product Ñ_{i} × ℝ; the action of π_{1}(M_{i}) on M̃_{i} preserves this product structure and so there is an induced action of π_{1}(M_{i}) on the ℝ factor by translations. Hence we get a homomorphism τ_{i} : π_{1}(M_{i}) → ℝ for each i. We may also define a class function MLS_{i} : π_{1}(M_{i}) → ℝ_{+} by taking the minimum of the displacement function for the induced action π_{1}(M_{i}) ↷ Ñ_{i}, i.e. MLS_{i}(g) = inf{dÑ_{i}(gx, x) | x ∈ Ñ_{i}}; this corresponds to the marked length spectrum of the nonpositively curved orbifold N_{i}.

Now suppose M and M′ are graph manifolds as above, and f : M → M′ is a homotopy equivalence. Embedded incompressible tori in Haken manifolds are determined up to isotopy by their fundamental groups up to conjugacy [Lau74], so we may assume after isotoping f that it is a homeomorphism which induces homeomorphisms f_{i} : M_{i} → M′_{i} from the Seifert components of M to the Seifert components of M′ (and hence isomorphisms on the corresponding fundamental groups). We may then use the maps f_{i} to compare the invariants τ_{i}, τ′_{i} and MLS_{i}, MLS′_{i}.

2Let M_{1} and M_{2} be closed surfaces with nonpositive curvature, and let N_{1} and N_{2} be the Riemannian products N_{i} := M_{i} × S^{1}. Suppose f_{0} : M_{1} → M_{2} is a homotopy equivalence, f := f_{0} × id_{S^{1}} : N_{1} → N_{2} is the corresponding map between the N_{i}’s, and f̂: Ñ_{1} → Ñ_{2} is a lift of f to a map between the universal covers. Then it turns out that f̂ extends continuously to up to the ideal boundary ∂_{∞}N_{1} if and only if f_{0} is homotopic to a homothety.

3Boundaries can even fail to be (non-equivariantly) homeomorphic: [CK] describes a pair of homeomorphic nonpositively curved 2-complexes whose universal covers have nonhomeomorphic boundary (see also [Wil]).

4The action G ↷ ∂_{∞}X is expansive if and only if G is hyperbolic.

Theorem 1.1. The following are equivalent:

1. The functions MLS_{i} and τ_{i} are preserved up to scale by f_{i}: for i = 1, …, k there are constants a_{i} and b_{i} so that MLS_{i} = a_{i}f^{∗}(MLS′_{i}) and τ_{i} = b_{i}f^{∗}(τ′_{i}).

2. Any lift f̃ : M̃ → M̃′ of f extends continuously to a map f̄ : M ∪ ∂_{∞}M → M′ ∪ ∂_{∞}M′ between the standard compactifications.

3. If f̃: M̃ → M̃′ is any lift of f, then f̃ maps geodesic rays to geodesic rays, up to uniform sublinear error: there is a function θ : ℝ_{+} → ℝ_{+} with lim_{r→∞} θ(r) = 0 so that if γ : [0, ∞) → M̃ is a unit speed geodesic ray, then there is a ray γ′ : [0, ∞) → M̃′ where d(f̃∘ γ(t), γ′([0, ∞))) < (1 + t)θ(t).

4. If we identify π_{1}(M) with π_{1}(M′) via f, then the induced boundary actions π_{1}(M) ↷ ∂_{∞}M̃ and π_{1}(M) ↷ ∂_{∞}M̃′ are equivariantly homeomorphic (by a unique equivariant homeomorphism).

If 1 holds and in addition the constants a_{i} and b_{i} are independent of i, then the unique equivariant homeomorphism ∂_{∞}M̃ → ∂_{∞}M̃′ in 4 is an isometry with respect to Tits metrics.

In general (see Lemma 6.2) the structure of π_{1}(M) forces the a_{i}’s and b_{i}’s in condition 1 to satisfy #{a_{1}, …, a_{k}, b_{1}, …, b_{k}} ≤ 2, and except in special circumstances they all coincide. The condition MLS_{i} = a_{i}f^{∗}(MLS′_{i}) means that the homotopy equivalence N_{i} → N′_{i} induced by f_{i} preserves the marked length spectrum of the nonpositively curved orbifolds up to the scale factor a_{i}. Although closed nonpositively curved surfaces with the same marked length spectrum are isometric by [Cro90, Ota90, CFF92], compact nonpositively curved surfaces with geodesic boundary can have the same marked length spectrum without being isometric:

Example 1.2. Let N be a pair of pants with a (constant curvature −1) hyperbolic metric where the boundary components are geodesics with length L, and let {c_{1}, c_{2}} ⊂ N be the fixed point set of the order 3 isometry of N. If L is sufficiently large (so that N looks like a bikini) then a closed geodesic in N cannot pass near {c_{1}, c_{2}}. This means that one can change the metric near {c_{1}, c_{2}} without disturbing the marked length spectrum of N. Note that one can modify this example slightly so that the metric is flat in a neighborhood of the boundary geodesics.

Suppose M is a nonpositively curved graph manifold with a Seifert component M_{i} isometric to N × S^{1}, where N is as in the example. One can change the metric on the N factor as in example 1.2 to get a Riemannian manifold M′ so that the conditions of Theorem 1.1 hold (with f = id), but M′ is not isometric to M.

In section 8 we give an example to show that the uniform sublinear divergence estimate in condition 3 cannot be improved to a bounded distance estimate as in the Gromov hyperbolic case.

We now sketch some of the main points in the proof of Theorem 1.1.

First consider a single nonpositively curved graph manifold M with geometric Seifert components M_{1}, …, M_{k}. The universal cover M̃_{i} is isometric to Ñ_{i} × ℝ – a nonpositively curved 3-manifold with a countable collection of totally geodesic boundary components isometric to 𝔼^{2}. The universal cover M̃ of M is tiled by a countable collection of copies of the universal covers M̃_{i} for i = 1, … k; we call these subsets vertex spaces. We refer to boundary components of vertex spaces as edge spaces. Two vertex spaces are either disjoint, or intersect along an edge space. Let T be the incidence graph for the collection of vertex spaces: T is the graph which has one vertex for each vertex space, and an edge joining two vertices whenever the corresponding vertex spaces intersect. T is isomorphic to the Bass-Serre tree of the graph of groups associated with the decomposition ⨿_{i}M_{i} → M (see section 2.5). If v ∈ V := V ertex(T) (resp. e ∈ E := Edge(T)) we will use the notation M̃_{v} (resp. M̃_{e}) for the vertex space (resp. edge space) associated with v (resp. e); and we let M̃_{v} ≃ Ñ_{v} × ℝ be the Riemannian product decomposition of M̃_{v}. It is not difficult to check (Lemma 3.23) that if G_{v} := Stabilizer(M̃_{v}) ⊂ G ≡ π_{1}(M), then the center Z(G_{v}) of G_{v} is isomorphic to ℤ, and the fixed point set of Z(G_{v}) in ∂_{∞}M̃ is just ∂_{∞}M̃_{v}; similarly, if M̃_{e} is an edge space then the fixed point set of ℤ^{2} ≃ G_{e} := Stabilizer(M̃_{e}) ⊂ G in ∂_{∞}M̃ is ∂_{∞}M̃_{e}.

Let p ∈ M̃ be an interior point of a vertex space, pick ξ ∈ ∂_{∞}M, and let \overline{pξ} denote the geodesic ray starting at p which is asymptotic to ξ. The ray \overline{pξ} encounters a (possibly finite) sequence of vertex and edge spaces called the itinerary of \overline{pξ}. The convexity of vertex and edge spaces forces the itinerary v_{0}, e_{1}, v_{1}, e_{2}, … of \overline{pξ} to be the sequence of successive vertices and edges of a geodesic segment or ray in T. In order to understand the rays with itinerary v_{0}, e_{1}, v_{1}, e_{2}, …, we construct a piecewise flat complex T – a template – in M̃ as follows. First let γ_{i} ⊂ M̃_{v_{i}} be a shortest geodesic from M̃_{e_{i}} to M̃_{e_{i+1}} for i > 0, and let γ_{0} be a shortest path from p ∈ M̃_{v_{0}} to M̃_{e_{1}}. For i ≥ 0 define S_{i} ⊂ M̃_{v_{i}} to be the flat strip which is the union of the geodesics in M̃_{v_{i}} which are parallel to the ℝ-factor of M̃_{v_{i}} and which pass through γ_{i}. We define T to be the union of the edge spaces {M̃_{e_{i}}} with the strips {S_{i}}; then T is a Hadamard space with respect to the induced path metric. A key technical step in the proof of Theorem 1.1 is Theorem 5.1, which shows that for any geodesic ray \overline{pζ} in the Hadamard space T , there is a unique geodesic ray \overline{pζ′} in M̃ with the property that for all x ∈ \overline{pζ},

dM̃(x, \overline{pζ′}) ≤ θ(d_{T} (x, p))(1 + d_{T} (x, p))

for some function θ : [0, ∞) → [0, ∞) with lim_{r→∞} θ(r) = 0 which is independent of the choice of itinerary. Using Theorem 5.1 one finds that the set of boundary points ζ ∈ ∂_{∞}M̃ for which the ray \overline{pζ} has a given infinite itinerary v_{0}, e_{1}, … is homeomorphic to the set boundary points ζ′ ∈ T so that the T -ray \overline{pζ′} passes through M̃_{e_{i}} for every i. One sees (Proposition 7.3) that the latter is either a single point or is homeomorphic to a closed interval, depending on the geometry of T (which depends, in turn, on the choice of itinerary and the geometry of M).

We now consider a second nonpositively curved 3-manifold M′, and use primes to denote the vertex spaces, edge spaces, etc for M′. Let f : M → M′ be a homeomorphism as in Theorem 1.1, and identify the deck groups G := π_{1}(M) ≃ π_{1}(M′) via a lift f̃ of f. Then f̃ maps vertex (resp. edge) spaces of M homeomorphically to vertex (resp. edge) spaces of M′, so we may use f̃ to identify the incidence tree T′ with T. Suppose ϕ : ∂_{∞}M̃ → ∂_{∞}M̃′ is a G-equivariant homeomorphism. Using the remarks about fixed point sets made above, it follows that ϕ(∂_{∞}M̃_{v}) = ∂_{∞}M̃′_{v} and ϕ(∂_{∞}M̃_{e}) = ∂_{∞}M̃′_{e} for every v ∈ V and every e ∈ E. Also, if p ∈ M̃_{v_{0}}, and v_{0}, e_{1}, v_{1}, e_{2}, … is an infinite itinerary, then ϕ(S) = S′ where S ⊂ ∂_{∞}M̃ and S′ ⊂ ∂_{∞}M̃′ are the subsets corresponding to the itinerary v_{0}, e_{1}, v_{1}, e_{2}, … (Corollary 5.28); in particular, either S and S′ are both points or they are both intervals. By considering all possible infinite itineraries and exploiting this correlation, we are able to see (section 7) that the invariants MLS_{i}, MLS′_{i} and τ_{i}, τ′_{i} must agree as in condition 1 of Theorem 1.1. Conversely, if condition 1 holds and p ∈ M̃, one shows (section 6) that for each itinerary the corresponding templates in M̃ and M̃′ have sufficiently similar geometry that their geodesics are “similar”; and this implies that f̃ extends to the compactifications as in 2 of Theorem 1.1.

Our main result generalizes Theorem 1.1 and applies to admissible groups, a class of (fundamental groups of) graphs of groups, see section 3.1 for the precise definition. When an admissible group G acts discretely and cocompactly on a Hadamard space X then we associate geometric data to each vertex group G_{v} ⊂ G consisting of a class function MLS_{v} : G_{v} → ℝ^{+} and a homomorphism τ_{v} : G_{v} → ℝ (see section 3.2).

Theorem 1.3. Let G ↷ X be a discrete, cocompact, isometric action of an admissible group on a Hadamard space X. Then for every vertex v, MLS_{v} and τ_{v} are determined up to scale factors a_{v} and b_{v} by the topological conjugacy class of the boundary action G ↷ ∂_{∞}X, and vice-versa. If G ↷ X′ is another such action, then the following are equivalent:

1. G ↷ X and G ↷ X′ have the same geometric data up to scale.

2. G-equivariant quasi-isometries X → X′ extend canonically to the compactifications X ∪ ∂_{∞}X → X′ ∪ ∂_{∞}X′.

3. The boundary actions G ↷ ∂_{∞}X and G ↷ ∂_{∞}X′ are G-equivariantly homeomorphic (by a unique^{5} G-equivariant homeomorphism).

4. If f : X → X′ is a G-equivariant quasi-isometry, then there is a function θ : ℝ_{+} → ℝ_{+} with lim_{r→∞} θ(r) = 0 so that for every unit speed geodesic ray γ : [0, ∞) → X there is a ray γ′ : [0, ∞) → X′ with d(f ∘ γ(t), γ′(t)) < (1 + t)θ(t).

Furthermore, if there is a single scale factor s so that MLS′_{v} = sMLS_{v} and τ′_{v} = sτ_{v} for every vertex v, then the unique G-equivariant homeomorphism ∂_{∞}X → ∂_{∞}X′ is an isometry with respect to the Tits metrics.

The authors proved theorem 1.3 while attempting to digest the negative answer to Gromov’s question about boundary actions. A key factor in our example was the (unanticipated) presence of intervals in the Tits boundary. After the examples and their properties had been announced, similar structure was found in other manifolds, [HS98]. The paper [BS] also contains some discussion of the Tits boundary of universal covers of nonpositively curved graph manifolds.

Open questions. The results in this paper raise a number of questions. First of all, for each group G one may ask for a generalization of Theorem 1.3, where the geometric data MLS_{v} and τ_{v} are replaced with suitable substitutes. Our methods actually yield more information about the behavior of geodesics than is stated in Theorem 1.3 alone. We are able to give a good description of all the geodesic rays in the Hadamard space X in terms of concrete geometric information; it seems likely that other classes of groups are amenable to a similar treatment. The fundamental groups of the real-analytic manifolds considered in [HS98] are natural candidates for this, as they have structure similar to graph manifold groups. Here are two other questions:

5It follows from the methods of [Bal95, III.3] that if G ↷ X is a cocompact isometric action on a Hadamard space, g ∈ G is an axial isometry, and γ_{g} ⊂ is an axis for g which does not bound a flat half plane, then the orbit of ∂_{∞}γ ⊂ ∂_{∞}X under the action G ↷ ∂_{∞}X is dense in ∂_{∞}X. Hence the set of points in ∂_{∞}X which are the unique attracting fixed point of some element of G is dense in ∂_{∞}X. Any G-equivariant homeomorphism ∂_{∞}X → ∂_{∞}X must fix this dense set pointwise, and must therefore be the identity. The uniqueness statement follows immediately from this.

1. What determines the (non-equivariant) homeomorphism type of ∂_{∞}X, when X is a Hadamard space with an action G ↷ X by an admissible group G?

2. Does part 4 of Theorem 1.3 have an analog where rays are replaced by complete geodesics? This seems within reach.

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1 Coarse geometry . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 Hadamard spaces . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.3 Groups acting on Hadamard spaces. . . . . . . . . . . . . . . . 9 2.4 Gromov hyperbolic groups and spaces . . . . . . . . . . . . . . 9 2.5 Graphs of groups and their Bass-Serre trees . . . . . . . . . . 13

3 Graphs of groups and the structure of Hadamard spaces on which they act . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.1 Admissible groups and actions . . . . . . . . . . . . . . . . . . 14 3.2 Vertex spaces, edge spaces, and geometric data for admissible

actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.3 Itineraries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4 Templates and the behavior of their geodesics . . . . . . . . . . . . . 22 4.1 Templates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 4.2 Templates associated with itineraries in T . . . . . . . . . . . 24

5 Shadowing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 5.1 Paths in a template which are close to a cluster of walls …. 29 5.2 Paths with small length distortion . . . . . . . . . . . . . . . . 31 5.3 The proof of Theorem 5.1 . . . . . . . . . . . . . . . . . . . . 34 5.4 Applications of Theorem 5.1 . . . . . . . . . . . . . . . . . . . 36

6 Geometric data and equivariant quasi-isometries . . . . . . . . . . . . 38 7 Recovering the geometric data from the action on the ideal boundary 42

7.1 ∂_{T}^{∞}T is either a point or an interval . . . . . . . . . . . . . . . 42 7.2 Self-similar Templates . . . . . . . . . . . . . . . . . . . . . . 43 7.3 Recovering the data . . . . . . . . . . . . . . . . . . . . . . . . 48 8 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

2. Preliminaries

2.1. Coarse geometry

Let X and X′ be metric spaces, and let Φ : X → X′ be a map.

Definition 2.1. 1. Φ is (L, A)-Lipschitz if for all x_{1}, x_{2} ∈ X,

d(Φ(x_{1}), Φ(x_{2})) ≤ Ld(x_{1}, x_{2}) + A.

Φ is coarse Lipschitz if it is (L, A)-Lipschitz for some L, A > 0.

2. Φ is an (L, A)-quasi-isometric embedding if it is (L, A)-Lipschitz and for all x_{1}, x_{2} ∈ X,

d(Φ(x_{1}), Φ(x_{2})) ≥ L^{−1}d(x_{1}, x_{2}) − A.

The constants (L, A) will often be suppressed. A quasi-geodesic (respectively segment/ray) is a quasi-isometric embedding Φ : ℝ → X (respectively Φ : [a, b] → X, Φ : [0, ∞) → X). We sometimes refer to the image of a quasi-geodesic as a quasigeodesic.

3. Φ is an (L, A)-quasi-isometry if it is an (L, A)-quasi-isometric embedding and for all x′ ∈ X′, d(x′, Φ(X)) < A.

4. Φ is a D-Hausdorff approximation if it is a (1, D)-quasi-isometry.

We will use the following well-known lemma:

Lemma 2.2. If G ↷ X is a discrete, cocompact, isometric action of a group G on a length space X, then there is a G-equivariant quasi-isometry Φ : Cayley(G) → X, where Cayley(G) is any Cayley graph of G.

2.2. Hadamard spaces

We refer the reader to [Bal95] for the material recalled here.

Geodesics and the boundary. Let X be a locally compact Hadamard space. If p, q ∈ X then \overline{pq} ⊂ X denotes the segment from p to q. If p, x, y ∈ X and p ∉ \{x, y\}, then ˜∠_{p}(x, y) (respectively ∠_{p}(x, y)) denotes the comparison angle (respectively angle) of the triangle Δpxy at p. We will use ∂_{∞}X to denote the set of asymptote classes of geodesic rays in X, with the cone topology. If p ∈ X and ξ ∈ ∂_{∞}X, then \overline{pξ} denotes the ray leaving p in the asymptote class of ξ. X̄ := X ∪ ∂_{∞}X denotes the usual compactification: a sequence x_{i} ∈ X̄ converges if and only if for any basepoint p ∈ X the sequence of geodesic segments/rays \overline{px_{i}} converges in the compact open topology. We denote the Tits angle between ξ_{1}, ξ_{2} ∈ ∂_{∞}X by ∠_{T} (ξ_{1}, ξ_{2}), and ∂_{T} X denotes the underlying set of ∂_{∞}X equipped with the Tits angle metric (which usually induces a topology different from the one defined above). The metric space ∂_{T} X is a CAT (1) space with respect to this metric. When ξ_{1}, ξ_{2} ∈ ∂_{T} X and ∠_{T} (ξ_{1}, ξ_{2}) < π then there is a segment between ξ_{1} and ξ_{2} in ∂_{T} X, which we denote by \overline{ξ_{1}ξ_{2}} ⊂ ∂_{T} X. This segment is the limit set in X̄ of any sequence of segments \overline{x_{1}^{k}x_{2}^{k}} where x_{i}^{k} tends to infinity along a ray asymptotic to ξ_{i}. We will not use the Tits path metric. We recall that ∂_{∞} and ∂_{T} behave nicely with respect to products: ∂_{∞}(X_{1} × X_{2}) = ∂_{∞}X_{1} ∘ ∂_{∞}X_{2} and ∂_{T} (X_{1} × X_{2}) = ∂_{T} X_{1} ∘ ∂_{T} X_{2} where in the first case ∘ represents the topological join and in the second the \frac{π}{2}-metric join. We will use this in the case where X_{2} = ℝ.

We will let N_{R}(C) be the closed metric tubular neighborhood of radius R of a subset C ⊂ X. A closed convex subset C ⊂ X is also a locally compact Hadamard space as is N_{R}(C), since it is also convex.

Standard comparison arguments show the following.

Lemma 2.3. Let X be a locally compact Hadamard space, and let C ⊂ X be a closed convex subset. Then for any R > 0, ξ ∈ ∂_{∞}N_{R}C, and z ∈ C we have \overline{zξ_{∞}} ⊂ C. In particular, ∂_{∞}C = ∂_{∞}N_{R}C.

One consequence is:

Lemma 2.4. Let X be a locally compact Hadamard space, and let C ⊂ X be a closed convex subset. If p ∈ X, ξ_{i} ∈ ∂_{∞}X, and ξ_{i} → ξ_{∞}, \overline{pξ_{i}} ∩ C ≠ ∅ for all i, then either \overline{pξ_{∞}} ∩ C ≠ ∅ or ξ_{∞} ∈ ∂_{∞}C.

Proof. Pick x_{i} ∈ \overline{pξ_{i}} ∩C. If lim inf d(x_{i}, p) < ∞ then a subsequence of x_{i} converges to x_{∞} ∈ C ∩ \overline{pξ_{∞}}. On the other hand, if lim inf d(x_{i}, p) = ∞ then for some subsequence \overline{px_{i}} → \overline{pξ_{∞}}. By the convexity of N_{R}(C), \overline{pξ_{∞}} ⊂ N_{R}(C) for R = d(p, C) and hence Lemma 2.3 yields the result.

Lemma 2.5. Let Φ : X → X′ be a quasi-isometric embedding, and assume there is a point x ∈ X and a function θ : ℝ_{+} → ℝ_{+} with lim_{r→∞} θ(r) = 0 so that for every y ∈ X, z ∈ \overline{xy}, we have

d_{X′}(Φ(z), \overline{Φ(x)Φ(y)}) ≤ (1 + d_{X}(z, x))θ(d_{X}(z, x)).  (2.6)

Then there is a unique extension Φ̄ : X̄ → X̄′ of Φ which is continuous at ∂_{∞}X, and ∂_{∞}Φ := Φ̄|_{∂_{∞}X} is a topological embedding.

Proof. Let Φ be an (L, A)−quasi isometric embedding, and ξ ∈ ∂_{∞}X, and y_{k} ∈ X be such that \overline{xy_{k}} converges to \overline{xξ}. By the convergence we can choose R_{k} → ∞ so that for all k ≥ n we have d(y_{k}, x) ≥ R_{n} and y_{kn} := \overline{xy_{k}} ∩ S(x, R_{n}) ⊂ N_{1}(\overline{xξ}). Note that the point on ξ closest to y_{kn} lies in ξ([R_{n} − 1, R_{n} + 1]), hence by the triangle inequality d(y_{kn}, y_{ln}) ≤ 4 for k, l ≥ n. Using (2.6), for every k ≥ n choose y′_{kn} ∈ \overline{Φ(x)Φ(y_{k})} with d_{X′}(y′_{kn}, Φ(y_{kn})) ≤ (1 + R_{n})θ(R_{n}). Then for every k, l ≥ n we have d_{X′}(Φ(y_{kn}), Φ(y_{ln})) ≤ 4L + A, and so d_{X′}(y′_{kn}, y′_{ln}) ≤ 2(1 + R_{n})θ(R_{n}) + 4L + A. This, along with the fact that d(Φ(x), y′_{kn}) ≥ L^{−1}R_{n} − A − (1 + R_{n})θ(R_{n}), forces ˜∠_{Φ(x)}(y′_{kn}, y′_{ln}) to zero as n → ∞. This in turn forces \overline{Φ(x)Φ(y_{k})} to converge to a ray \overline{Φ(x)Φ(ξ)} since for each R > 0 we have for large enough k that the sequence \{\overline{Φ(x)Φ(y_{k})} ∩ S(Φ(x), R)\} is Cauchy and hence converges. This proves that Φ has a unique extension Φ̄ : X̄ → X̄′ which is continuous at ∂_{∞}X. The map ∂_{∞}Φ := Φ̄|_{∂_{∞}X} clearly has the property that for all ξ ∈ ∂_{∞}X and all y ∈ xξ,

d(Φ(y), \overline{Φ(x)∂_{∞}Φ(ξ)}) ≤ (1 + d(x, y))θ(d(x, y)).  (2.7)

When ξ_{1}, ξ_{2} ∈ ∂_{∞}X are distinct, the rays \overline{xξ_{i}} diverge linearly, and hence Φ(\overline{xξ_{1}}) and Φ(\overline{xξ_{2}}) diverge linearly since Φ is a quasi isometric embedding. Now if ∂_{∞}Φ(ξ_{1}) = ∂_{∞}Φ(ξ_{2}) then (2.7) would imply that Φ(\overline{xξ_{1}}) and Φ(\overline{xξ_{2}}) would each diverge sublinearly from \overline{Φ(x)∂_{∞}Φ(ξ_{1})} and hence diverge sublinearly from each other. Thus we conclude that ∂_{∞}Φ(ξ_{1}) ≠ ∂_{∞}Φ(ξ_{2}).

2.3. Groups acting on Hadamard spaces.

Let X be a Hadamard space. We denote the displacement function of an isometry g : X → X by d_{g}, and the infimum of d_{g} by δ_{g}. When g is axial, we let Minset(g) denote the convex subset where d_{g} attains its minimum. We recall that Minset(g) splits as a metric product C × ℝ where C is convex and g acts trivially on the C factor and by translation on the ℝ factor.

Let G ↷ X be a discrete, cocompact, isometric action of a group G on a Hadamard space X. If H ⊂ G is a subgroup isomorphic to ℤ^{k}, we let Minset(H) := ∩_{h∈H}Minset(h). We recall that Minset(H) = ∩_{h∈S}Minset(h) for any generating set S ⊂ H, and that Minset(H) splits isometrically as a metric product C × 𝔼^{k} so that H acts trivially on the C factor, and as a translation lattice on the 𝔼^{k} factor. The centralizer Z(H, G) of H in G preserves δ_{h} for every h ∈ H, and hence also Minset(H). If S ⊂ H is a finite generating set, then the function ∑_{h∈S} δ_{h} : X → ℝ descends to a proper function on X/Z(H, G); in particular, Minset(H)/Z(H, G) is compact.

2.4. Gromov hyperbolic groups and spaces

For background on the material in this section see [Gro87], [GdlH90], and [CDP90]. Some standard facts that we will use: If a Gromov hyperbolic group G acts cocompactly on a Hadamard space X then (since X is then quasi-isometric to Cayley(G) and Gromov hyperbolicity is a quasi isometry invariant) X is Gromov hyperbolic (i.e. δ-hyperbolic for some δ). Further ∂_{∞}X is homeomorphic to ∂_{∞}G, and all infinite order elements g ∈ G are axial. The Tits metric ∂_{T} X is the discrete metric with any two distinct points having distance π.

In this section we make use of the Morse lemma for quasi-geodesic segments (see [Gro87, CDP90]):

Lemma 2.8. (Morse Lemma) Given δ > 0, L > 0 and A ≥ 0 there is a constant C = C(δ, L, A) such that if γ_{1} and γ_{2} are (L, A)-quasi-geodesic segments with the same endpoints sitting in a δ-hyperbolic space, then their Hausdorff distance satisfies d_{H}(γ_{1}, γ_{2}) < C.

Two geodesics γ_{1} and γ_{2} in a Hadamard space X are parallel if they stay a bounded distance apart. The parallel set P (γ) ⊂ X of a geodesic γ is the union of all geodesics parallel to γ. By the flat strip theorem, P (γ) is a convex subset of X, and is isometric to C_{γ} ×ℝ where C_{γ} ⊂ X is convex. A bounded convex set C always contains a unique circumcenter: the center of the smallest metric ball containing C.

Lemma 2.9. Let X be a δ-hyperbolic Hadamard space. Then

1. If γ ⊂ X is a geodesic and P (γ) ≃ C_{γ}×ℝ is its parallel set, then Diam(C_{γ}) < δ. In particular P (γ) contains a canonical geodesic z × ℝ ⊂ C_{γ} × ℝ where z ∈ C_{γ} is the circumcenter of C_{γ}.

2. If γ_{1}, γ_{2} ⊂ X are geodesics, x_{i} ∈ γ_{i}, then γ_{1} ∪ \overline{x_{1}x_{2}} ∪ γ_{2} is 2δ-quasi-convex^{6}.

3. Suppose γ_{1}, γ_{2} ⊂ X are geodesics with ∂_{∞}γ_{1} ∩ ∂_{∞}γ_{2} = ∅, and let η a minimal geodesic segment between γ_{1} and γ_{2}. Then any geodesic segment running from γ_{1} to γ_{2} will pass within distance D = D(γ_{1}, γ_{2}) of both endpoints of η; when d(γ_{1}, γ_{2}) > 4δ then we may take D = 2δ.

Proof. 1 follows from the fact that a δ-hyperbolic Euclidean strip has width at most δ. 2 and 3 follow from repeated application of the δ-thinness property of geodesic triangles.

In the following lemma is a slight variation on results from [Gro87]. It shows that discrete isometric actions on Gromov hyperbolic spaces behave like free group actions on trees.

Lemma 2.10. Let X be a δ-hyperbolic Hadamard space, and let ⋆ ∈ X. Suppose (g_{i})_{i∈ℤ} is a periodic sequence of axial isometries of X with period k (i.e. g_{i+k} = g_{i} for all i; and in particular g_{0} = g_{k} and g_{−1} = g_{k−1}), and let the attracting (respectively repelling) fixed point of g_{i} be ξ_{i}^{+} ∈ ∂_{∞}X (respectively ξ_{i}^{−} ∈ ∂_{∞}X). If for every i we have ξ_{i}^{−} ≠ ξ_{i+1}^{+}, then there are constants L, A, N, and D with the following property.

1. If (m_{i})_{i∈ℤ} is a sequence with m_{i} > N, then the broken geodesic with vertices

…, v_{−2} = (g_{−1}^{−m_{−1}}g_{−2}^{−m_{−2}})(⋆), v_{−1} = g_{−1}^{−m_{−1}}(⋆), v_{0} = ⋆, v_{1} = g_{0}^{m_{0}}(⋆), v_{2} = (g_{0}^{m_{0}}g_{1}^{m_{1}})(⋆), … (2.11)

is an (L, A) quasi-geodesic, and

|d(v_{i}, v_{i+l}) − ∑_{j=i}^{i+l−1} m_{j}δ_{g_{j}}| < lD  (2.12)

2. If (m_{i})_{i∈ℤ} is a sequence with m_{i} > N and period k, then g := g_{0}^{m_{0}} … g_{k−1}^{m_{k−1}} is an axial isometry with an axis γ within Hausdorff distance D of the g-invariant broken geodesic with vertices (2.11), and the minimal displacement of g satisfies

|δ_{g} − (m_{0}δ_{g_{0}} + … + m_{k−1}δ_{g_{k−1}})| < D.  (2.13)

Furthermore, as m_{0} → ∞ (respectively m_{k−1} → ∞), the attracting (respectively repelling) fixed point of g tends to ξ_{0}^{+} (respectively ξ_{k−1}^{−}).

Proof. Since ξ_{i}^{−} ≠ ξ_{i+1}^{+} for all i ∈ ℤ, there are constants L_{1}, A_{1}, and N_{1}, so that when m_{±} > N_{1} then for any i the broken geodesic with vertices g_{i}^{−m_{−}}(⋆), ⋆, g_{i+1}^{m_{+}}⋆ is an (L_{1}, A_{1}) quasi-geodesic segment. Let (m_{i})_{i∈ℤ} be a sequence, and let η : ℝ → X be the broken geodesic with vertices (2.11). By the local characterization of quasi-geodesics given in [CDP90, Chapitre 3], we get constants L = L(L_{1}, A_{1}, δ), A = A(L_{1}, A_{1}, δ), and N_{2} = N_{2}(L_{1}, A_{1}, δ, {g_{i}}_{i∈ℤ}) ≥ N_{1} so that η is an (L, A) quasi-geodesic provided m_{i} ≥ N_{2} for all i.

6A subset Z ⊂ X is C-quasi-convex if for all x, y ∈ Z we have \overline{xy} ⊂ N_{C}(Z).

We now assume that m_{i} ≥ N_{2} for all i. By the Morse lemma there is a D_{1} = D_{1}(L, A, δ) so that there is a geodesic at Hausdorff distance at most D_{1} from η(ℝ), and any geodesic γ ⊂ X with ∂_{∞}γ = ∂_{∞}η has Hausdorff distance at most D_{1} from η. Fix such a geodesic γ ⊂ X, and for each i ∈ ℤ let w_{i} ∈ γ be the point in γ nearest v_{i}. By the triangle inequality we have

|d(v_{i}, v_{i+1}) − d(w_{i}, w_{i+1})| ≤ 2D_{1}.  (2.14)

Choose c_{1} = c_{1}(⋆, {g_{i}}) so that the distance from ⋆ to the nearest axis of g_{i} is less than c_{1}; then for all i ∈ ℤ

|d(v_{i}, v_{i+1}) − m_{i}δ_{g_{i}}| = |d(⋆, g_{i}^{m_{i}}(⋆)) − m_{i}δ_{g_{i}}| < 2c_{1}.  (2.15)

Since for each i, the broken segment with vertices v_{i−1}, v_{i}, v_{i+1} is an (L_{1}, A_{1}) quasigeodesic, the Morse Lemma gives

d(v_{i−1}, v_{i+1}) ≥ d(v_{i−1}, v_{i}) + d(v_{i}, v_{i+1}) − 2D_{1}.  (2.16)

This gives

d(w_{i−1}, w_{i+1}) ≥ d(w_{i−1}, w_{i}) + d(w_{i}, w_{i+1}) − 8D_{1}.  (2.17)

Therefore there is an N = N(⋆, {g_{i}}) ≥ N_{2} so that if m_{i} ≥ N then w_{i} lies between w_{i−1} and w_{i+1} for all i. So when m_{i} ≥ N we have

|d(v_{i}, v_{i+l}) − ∑_{j=i}^{j+l−1} m_{j}δ_{g_{j}}| ≤ 2D_{1} + |d(w_{i}, w_{i+l}) − ∑_{j=i}^{j+l−1} m_{j}δ_{g_{j}}|

≤ 2D_{1} + ∑_{j=i}^{j+l−1} |d(w_{j}, w_{j+1}) − m_{j}δ_{g_{j}}| ≤ 2D_{1}(l + 1) + 2lc_{1}.  (2.18)

We now set D := (2k + 4)D_{1} + 2kc_{1}, and note that we have proved 1. When the sequence (m_{i}) has period k, m_{i} ≥ N for all i, and g := g_{0}^{m_{0}} … g_{k}^{m_{k−1}}, then we may take γ to be an axis for g. We have δ_{g} = d(w_{0}, w_{k}), and (2.13) follows from (2.18). The last assertion follows immediately from the fact that as m_{0} → ∞ and m_{k−1} → ∞, the segments \overline{⋆g_{0}^{m_{0}}(⋆)} and \overline{⋆g_{k−1}^{−m_{k−1}}(⋆)} converge to the rays \overline{⋆ξ_{0}^{+}} and \overline{⋆ξ_{k−1}^{−}} respectively.

Lemma 2.19. Let G ↷^{ρ} X be a discrete, cocompact isometric action of a hyperbolic group G on a Hadamard space X. There is a constant D = D(δ, ρ) so that for every x_{1}, x_{2} ∈ X there is a g ∈ G and an axis γ for g with d(x_{i}, γ) < D for i = 1, 2, and d(g(x_{1}), x_{2}) < D.

Proof. If G is elementary, then either G is finite (in which case the result holds trivially) or there is a hyperbolic element g ∈ G with an axis γ so that X = N_{R}(γ) for some R; this implies 2 in this case. So we may assume that G is nonelementary, and hence G does not fix any ξ ∈ ∂_{∞}X.

Pick ⋆ ∈ X and a finite generating set Σ ⊂ G. Fix σ_{0} ∈ Σ, let Σ′ = {σ_{0}}∪{σσ_{0}|σ ∈ Σ}, and let C(Σ′) = min{d(⋆, σ′(⋆))|σ′ ∈ Σ}.

We note that by the cocompactness of the action it is sufficient to prove the theorem when x_{1} is ⋆; for then (with a larger D) if g_{1}(x_{i}) is near ⋆ (within the diameter of the fundamental domain) and g is the solution for ⋆ and g_{1}(x_{2}) then g_{1}^{−1}gg_{1} works for x_{1} and x_{2} (since g_{1}^{−1}(γ_{g}) is an axis for g_{1}^{−1}gg_{1}). Claim. There are constants L_{1}, A_{1} such that for all g ∈ G, there is a σ′ ∈ Σ′ so that the broken geodesic with vertices (gσ′)^{−1})(⋆), ⋆, (gσ′)(⋆) is an (L_{1}, A_{1})-quasi-geodesic. Proof of claim. If not, there is a sequence g_{k} ∈ G with d(g_{k}(⋆), ⋆) → ∞, so that for every σ′ ∈ Σ′ the broken geodesic with vertices (g_{k}σ′)^{−1})(⋆), ⋆, (g_{k}σ′)(⋆) is not a (k, k)-quasi-geodesic. This clearly implies that after passing to a subsequence (which works for all σ′), the segments \overline{⋆[(g_{k}σ′)(⋆)]}, \overline{⋆[(g_{k}σ′)^{−1}(⋆)]} converge to some ray \overline{⋆ξ(σ′)}. But since d(g_{k}(⋆), (g_{k}σ′)(⋆)) ≤ C(Σ′) we see (g_{k})(⋆) converges to ξ(σ′), so ξ(σ′) = ξ is independent of σ′. The fact that (g_{k}σ′)^{−1}(⋆) converges to ξ tells us (by applying σ′ to the sequence) that g_{k}^{−1}(⋆) converges to σ′(ξ) = ξ′ which is again independent of the choice of σ′. Thus σ′(ξ) = ξ′ for all σ′ ∈ Σ, and hence for every σ ∈ Σ, σ(ξ′) = (σσ_{0})σ_{0}^{−1}(ξ′) = ξ′, which is a contradiction. Proof of Lemma 2.19 continued. By the claim and an application of [CDP90, Chapitre 3] as in the proof of Lemma 2.10, we see that there are constants L, A, and D_{1} so that if g ∈ G and d(g(⋆), ⋆) > D_{1}, then there is a σ′ ∈ Σ′ so that the broken geodesic with i^{th} vertex (gσ′)^{i}(⋆) is an (L, A)-quasi-geodesic. By the Morse Lemma and Lemma 2.9, (gσ′) has an axis at distance < D_{2}(L, A, δ) from ⋆ and (gσ′)(⋆). Since d((gσ′)(⋆), g(⋆)) = d(σ′(⋆), ⋆) ≤ C(Σ′), the lemma clearly follows.

Lemma 2.20. Consider two discrete, cocompact, isometric actions G ↷^{ρ} X, G ↷^{ρ′} X′ where X and X′ are δ-hyperbolic Hadamard spaces. Assume that the minimum displacement of any g ∈ G in X is the same as the minimal displacement in X′. Then any G-equivariant (L, A)-quasi-isometry Φ : X → X′ maps unit speed geodesics γ to within D = D(L, A, δ, ρ, ρ′) of a unit speed geodesic γ′, that is d(Φ(γ(t)), γ′(t)) ≤ D.

Proof. By the Morse lemma on quasi-geodesics, it suffices to show that Φ is a D_{1} = D_{1}(L, A, δ, ρ, ρ′) Hausdorff approximation. Let D_{5} = max{D_{4}(δ, ρ), D_{4}(δ, ρ′)} where the D_{4}’s come from Lemma 2.19. For x_{1}, x_{2} ∈ X (resp. Φ(x_{1}), Φ(x_{2}) ∈ X′) let g ∈ G (resp g′ ∈ G) be the elements guaranteed by Lemma 2.19. Since x_{1} is D_{5} close to an axis of g we know that 2D_{5} + δ_{g} ≥ d(x_{1}, g(x_{1})) ≥ δ_{g} and hence 3D_{5} + δ_{g} ≥ d(x_{1}, x_{2}) ≥ δ_{g} − D_{5}. Now

d(Φ(x_{1}), Φ(x_{2})) ≥ d(Φ(x_{1}), Φ(g(x_{1}))) − d(Φ(g(x_{1})), Φ(x_{2})) ≥
≥ δ_{g} − LD_{5} − A ≥ d(x_{1}, x_{2}) − (L + 3)D_{5} − A

and similarly

d(x_{1}, x_{2}) ≥ d(x_{1}, g′(x_{1})) − d(g′(x_{1}), x_{2}) ≥
≥ δ_{g′} − LD_{5} − A ≥ d(Φ(x_{1}), Φ(x_{2})) − (L + 3)D_{5} − A.

Hence we can take D_{1} = (L + 3)D_{5} + A.

Lemma 2.21. Let G ↷^{ρ} X be a discrete, cocompact isometric action on a δ-Hyperbolic Hadamard space X.

1. If {γ_{g}|g ∈ S ⊂ G} is a collection of distinct axis of distinct elements g ∈ G such that {δ_{g}} is bounded then {γ_{g}} forms a discrete set of geodesics.

2. If two axial elements g_{1}, g_{2} ∈ G have a common fixed point in ∂_{∞}X then they have a common axis (i.e. they have both fixed points in common).

Proof. Assume some sequence γ_{g_{i}} converges to a geodesic γ and let ⋆ ∈ γ then the boundedness of {δ_{g_{i}}} says that there is a C such that d(⋆, g_{i}(⋆)) < C but this cannot be true for infinitely many distinct g_{i}.

To see the second statement we can assume that the attracting and repelling fixed points satisfy ξ_{1}^{+} = ξ_{2}^{+} and ξ_{1}^{−} ≠ ξ_{2}^{−} (the other cases are similar). In this case {g_{1}^{−k}g_{2}g_{1}^{k}} are distinct since they have distinct fixed point sets {ξ_{1}^{+}, g_{1}^{−k}(ξ_{2}^{−})} in ∂_{∞}X while the axes converge (after taking a subsequence) to an axis of g_{k}. But again there is a C such that d(⋆, g_{1}^{−k}g_{2}g_{1}^{k}(⋆)) < C giving the desired contradiction.

2.5. Graphs of groups and their Bass-Serre trees

For the remainder of the paper, all group actions on simplicial trees will be assumed to be simplicial actions which do not invert edges, and geodesic segments/rays in simplicial trees will be unions of edges.

References for the material in this section are [Ser80, SW79, DD89].

Definition 2.22. A graph of groups is a connected graph G together with a group G_{σ} labeling each σ ∈ V ertex(G) ∪ Edge(G), and a monomorphism G_{e} → G_{v} for each pair (e, v) consisting of an oriented edge e entering a vertex v. An isomorphism of two graphs of groups is an isomorphism of labeled graphs which is compatible with edge monomorphisms.

Let G ↷^{ρ} T be an action of a group G on a simplicial tree T. We can define an associated graph of groups G as follows. We let the graph underlying G be G/T. For each σ ∈ V ertex(G) ∪ Edge(G) we may label σ with the stabilizer of a lift σ̂ ⊂ T of σ. An for each pair (e, v), where e ⊂ T/G is an oriented edge with terminus v ∈ T/G, we can define an edge monomorphism G_{e} → G_{v} by composing the inclusion G_{e} := Gê → G_{gv̂} (gv̂ ∈ T is the terminus of ê) with the isomorphism G_{gv̂} → Gv̂ = G_{v} induced by conjugation by g^{−1}. We refer to this as the graph of groups associated with the action G ↷ T.

Lemma 2.23. If G is a graph of groups, then there is a group G, a simplicial tree T, and an action G ↷^{ρ} T so that:

1. If Ḡ is the graph of groups associated with the action G ↷^{ρ} T, then Ḡ ≃ G.

2. If G′ ↷^{ρ′} T′ is another action on a simplicial tree satisfying 1, then there is an isomorphism G′ ≃ G so that the actions ρ and ρ′ become simplicially isomorphic.

The (isomorphism class of the) group G is the fundamental group of G, and the tree T (or really the action G ↷^{ρ} T) is called the Bass-Serre tree of G. We note that if v is a vertex of T and G_{v} is its stabilizer, then the G_{v}-orbits of Link(v) correspond bijectively to the elements of Link(v̄) where v̄ ∈ G is the corresponding vertex of G/T ≃ G, and the stabilizer of ξ ∈ Link(v) is just G_{e} where e is the edge associated with ξ.

Let G be a graph of groups and let ē be an edge of G with endpoints v̄_{1} and v̄_{2}. We let G′ be the graph of groups determined by ē. The fundamental group of G′ is a free product with amalgamation if ē is embedded in G and an HNN extension if ē is a loop. Choose a lift e = \overline{v_{1}v_{2}} ⊂ T of ē to the Bass-Serre tree T. We may identify Gv̄_{i} with G_{v_{i}} and Gē with G_{e} in a fashion compatible with the edge inclusions Gē → Gv̄_{i}, G_{e} → G_{v_{i}}. When ē is a loop we may choose t ∈ G = π_{1}(G) so that t(v_{1}) = v_{2} and the

composition G_{e} → G_{v_{1}} ⟶^{t(·)t^{−1}} G_{v_{2}} agrees with the edge monomorphism G_{e} → G_{v_{2}}. Set G′ := ⟨G_{v_{1}}, G_{v_{2}}⟩ when ē is embedded and set G′ := ⟨G_{v_{1}}, t⟩ when ē is a loop. Then the orbit T′ := G′(e) ⊂ T is a G′-invariant subtree of T, and the action G′ ↷ T′ is the Bass-Serre action for G′. When ē is embedded we choose subsets Σ_{i} ⊂ G_{v_{i}} which intersect each right coset of G_{e} exactly once; then any g ∈ G′ can be written uniquely in the form

s_{1} … s_{k}r  (2.24)

where r ∈ G_{e}, s_{i} ∉ G_{e}, and the s_{i}’s belong alternately to Σ_{1} and Σ_{2}. The combinatorial distance from the edge g(e) to e is k and e_{i} = s_{1} … s_{i}(e) is the sequence of edges along the path from e to g(e). When ē is a loop, we choose a cross-section Σ_{1} ⊂ G_{v_{1}} (respectively Σ_{−1}) of the right cosets of G_{e} (respectively t^{−1}G_{e}t). Then any g ∈ G′ can be written uniquely in the form

s_{1}t^{ϵ_{1}}s_{2}t^{ϵ_{2}} … s_{k}t^{ϵ_{k}}r  (2.25)

where for i = 1, …, k, ϵ_{i} = ±1, s_{i} ∈ Σ_{ϵ_{i}}, r ∈ G_{v_{1}}, and if s_{i} ∈ G_{e} then ϵ_{i−1} = −ϵ_{i}.

3. Graphs of groups and the structure of Hadamard spaces on which they act

3.1. Admissible groups and actions

Definition 3.1. A graph of groups G is admissible if 1. G is a finite graph with at least one edge.

2. Each vertex group Ḡ_{v} has center Z(Ḡ_{v}) ≃ ℤ, H̄_{v} := Ḡ_{v}/Z(Ḡ_{v}) is a nonelementary hyperbolic group, and every edge subgroup Ḡ_{e} is isomorphic to ℤ^{2}.

3. Let e_{1}, e_{2} be distinct directed edges entering a vertex v, and for i = 1, 2 let K_{i} ⊂ Ḡ_{v} be the image of the edge homomorphism Ḡ_{e_{i}} → Ḡ_{v}. Then for every g ∈ Ḡ_{v}, gK_{1}g^{−1} is not commensurable with K_{2}, and for every g ∈ Ḡ_{v} − K_{i}, gK_{i}g^{−1} is not commensurable with K_{i}.

4. For every edge group Ḡ_{e}, if α_{i} : Ḡ_{e} → Ḡ_{v_{i}} are the edge monomorphisms, then the subgroup generated by α_{1}^{−1}(Z(Ḡ_{v_{1}})) and α_{2}^{−1}(Z(Ḡ_{v_{2}})) has finite index in G_{e} ≃ ℤ^{2}. A group G is admissible if it is the fundamental group of an admissible graph of groups.

Let G be the fundamental group of an admissible graph of groups G, and let G ↷ T be the action of G on the associated Bass-Serre tree. We let V := V ertex(T) and E := Edge(T) denote the vertex and edge sets of T, and when σ ∈ V ∪ E we let G_{σ} ⊂ G denote corresponding stabilizer. Properties 1-4 of definition 3.1 imply :

Lemma 3.2. 1. T is an unbounded tree with infinite valence at each vertex, and G acts on T with quotient G.

2. Each vertex group G_{v} has center Z(G_{v}) ≃ ℤ, H_{v} := G_{v}/Z(G_{v}) is a nonelementary hyperbolic group, and every edge subgroup G_{e} is isomorphic to ℤ^{2}.

3. If e_{1}, e_{2} are distinct edges emanating from v ∈ V, then G_{e_{1}} is not commensurable with G_{e_{2}}. In particular, Z(G_{v}) ⊂ G_{e_{i}} since g ∈ Z(G_{v}) implies G_{e_{i}} = gG_{e_{i}}g^{−1} = G_{ge_{i}} forcing ge_{i} = e_{i}, i.e. g ∈ G_{e_{i}}.

4. If e ∈ E has endpoints v_{1}, v_{2} ∈ V, then (Z(G_{v_{1}}) ∪ Z(G_{v_{2}})) ⊂ G_{e} generates a finite index subgroup of G_{e}.

Most of the time we will work with the action G ↷ T and ignore the graph of groups that produced it.

Examples of admissible groups:

1. (Graph manifolds) Let M be a 3-dimensional nonpositively curved graph manifold as in Theorem 1.1, and let M_{i}, i = 1, … k be the geometric Seifert components of M. Let G be the graph of groups which has one vertex labeled with π_{1}(M_{i}) for each i, and an edge labeled by ℤ^{2} for each pair of totally geodesic boundary tori in the disjoint union ∪_{i}M_{i} which are glued to form M. The edge monomorphisms come from the two different embeddings of a gluing torus into Seifert components.

2. (Torus complexes) Let T_{0}, T_{1}, T_{2} be flat two-dimensional tori. For i = 1, 2, we choose primitive closed geodesics a_{i} ⊂ T_{0} and b_{i} ⊂ T_{i} with length(a_{i}) = length(b_{i}), and we glue T_{i} to T_{0} by identifying a_{i} with b_{i} isometrically. We assume that a_{1} and a_{2} lie in distinct free homotopy classes, and intersect at an angle α ∈ (0, \frac{π}{2}]. Let G be the graph of groups associated with the decomposition (T_{0} ∪ T_{1}) ⨿ (T_{0} ∪ T_{1}) → ∪T_{i}. Note that T_{0} ∪ T_{1} is homeomorphic to S^{1} × (S^{1} ∧ S^{1}), so π_{1}(T_{0} ∪ T_{i}) = ℤ × F_{2} where F_{2} is the free group on two generators.

Lemma 3.3. 1. If e_{1}, e_{2} ∈ E are distinct edges incident to v ∈ V, then Z(G_{v}) ≃ ℤ is a finite index subgroup of G_{e_{1}} ∩ G_{e_{2}}. In particular G_{e_{1}} ∩ G_{e_{2}} ≃ ℤ.

2. If v_{1}, v_{2} ∈ V are the endpoints of an edge e ∈ E, then Z(G_{v_{1}}) ∩ Z(G_{v_{2}}) = {id}.

Proof. Note that G_{e_{1}} ∩ G_{e_{2}} has infinite index in each G_{e_{i}}, for otherwise the G_{e_{i}} ≃ ℤ^{2} would be commensurable, contradicting 3 of Lemma 3.2. Thus G_{e_{1}} ∩ G_{e_{2}} ≃ ℤ. Also by 3 of Lemma 3.2 we have Z(G_{v}) ⊂ G_{e_{1}} ∩G_{e_{2}}, so both are rank 1 free abelian groups and 1 follows.

2 follows immediately from 4 of Lemma 3.2, since the Z(G_{v_{i}}) are rank 1 subgroups of G_{e} ≃ ℤ^{2} which generate a finite index subgroup of G_{e}.

Lemma 3.4. If v_{1}, v_{2} ∈ V, then 1. If d(v_{1}, v_{2}) > 2 then G_{v_{1}} ∩ G_{v_{2}} = {id}.

2. If d(v_{1}, v_{2}) = 2 and v ∈ V is the vertex between them, then Z(G_{v}) is a finite index subgroup of G_{v_{1}} ∩ G_{v_{2}}.

3. If d(v_{1}, v_{2}) = 1, then G_{v_{1}} ∩ G_{v_{2}} = G_{e} where e = \overline{v_{1}v_{2}}.

Proof. We will prove the assertions in reverse order. Part 3 is immediate since the action G ↷ T does not invert edges, see section 2.5. To prove 2 we let e_{i} be the edge between v_{i} and v. Clearly G_{v_{1}} ∩ G_{v_{2}} = G_{e_{1}} ∩ G_{e_{2}}, which by Lemma 3.3 contains Z(G_{v}) as a subgroup of finite index. To prove 1, let e_{1}, e, e_{2} ∈ E be three consecutive edges of the segment \overline{v_{1}v_{2}}, and let w_{1}, w_{2} ∈ V be the endpoints of e. Then by 1 of Lemma 3.3, Z(G_{w_{i}}) (≃ ℤ) has finite index in G_{e_{i}} ∩ G_{e} (≃ ℤ), so we have (G_{e_{1}} ∩ G_{e}) ∩ (G_{e} ∩ G_{e_{2}}) = ∅ since otherwise Z(G_{w_{1}}) ∪ Z(G_{w_{2}}) would generate a cyclic group in G_{e} (≃ ℤ^{2}) contradicting 4 of Lemma 3.2. Since

G_{v_{1}} ∩ G_{v_{2}} ⊂ (G_{e_{1}} ∩ G_{e}) ∩ (G_{e} ∩ G_{e_{2}})

1 follows.

Lemma 3.5. For v ∈ V, the fixed point set of Z(G_{v}) is the closed star \overline{Star(v)}. For e ∈ E, the fixed point set of G_{e} is e. In particular, for any σ ∈ V ∪ E, G_{σ} leaves no point in ∂_{∞}T fixed. Further, if \overline{Star(v)} is invariant under Z(G_{σ}) then σ ∈ \overline{Star(v)}.

Proof. Fix a vertex v. Since Z(G_{v}) ⊂ G_{e} for all e in the star it is clear that the closed star is in the fixed point set of Z(G_{v}). On the other hand 1 of Lemma 3.4 says we need only consider vertices v_{1} such that d(v, v_{1}) = 2. Let w be the vertex between v and v_{1}. Now 2 of Lemma 3.4 says that Z(G_{w}) is a finite index subgroup of G_{v} ∩ G_{v_{1}}, while 2 of Lemma 3.3 says that Z(G_{v}) ∩ Z(G_{w}) = ∅. Thus Z(G_{v}) ∩ G_{v_{1}} = ∅ and the first statement follows.

For an edge e = \overline{v_{1}v_{2}} the first part (since G_{v_{i}} ⊂ G_{e}) says that the fixed point set of G_{e} is contained in \overline{Star(v_{1})} ∩ \overline{Star(v_{2})} = e. So the second statement follows.

The last two statements follow from the first two.

Lemma 3.6. If v ∈ V then the centralizer of Z(G_{v}) in G is just G_{v}. If e ∈ E the centralizer of G_{e} in G is G_{e}.

Proof. By Lemma 3.5, the fixed point set of Z(G_{v}) in T is just the closed star of v in T. Hence any g ∈ G which commutes with Z(G_{v}) must take the star of v to itself and hence fix v.

If e ∈ E and e = \overline{v_{1}v_{2}}, then a finite index subgroup G_{e} is generated by Z(G_{v_{1}}) ∪ Z(G_{v_{2}}). So the centralizer of G_{e} in G is a subgroup of the intersection of the centralizers of Z(G_{v_{1}}) and Z(G_{v_{2}}), i.e. G_{v_{1}} ∩ G_{v_{2}} which is G_{e} itself.

Lemma 3.7. (Uniqueness of decomposition) Let G ↷ T and G′ ↷ T′ be the Bass-Serre actions associated with two admissible graphs of groups, and suppose G →^{ϕ} G′ is an isomorphism. Then after identifying G with G′ via ϕ, the trees T and T′ become G-equivariantly isomorphic.

Proof. We will use primes to denote the vertex and edge set of T′. Pick v ∈ V.

Claim: G_{v} fixes a unique vertex in T′. Let g ∈ Z(G_{v}) be a generator. If F ix(g, T′) is empty, then g translates a unique geodesic γ ⊂ T′, and since g ∈ Z(G_{v}) the whole vertex group G_{v} must preserve γ, and act on it by translations. The signed translation distance yields a homomorphism G_{v} → ℤ with nontrivial kernel. But then Ker(G_{v} → ℤ) fixes γ pointwise, which contradicts 1 of Lemma 3.4. Consequently F ix(g, T′) is nonempty, and by 1 of Lemma 3.4 this is a subcomplex of T′ with diameter at most 2. So G_{v} must fix the center of F ix(g, T′). It can fix nothing more, since no edge stabilizer can contain the nonabelian G_{v}. Thus we have proved the claim.

Now consider the G-equivariant map f : V → V′ which assigns to each v ∈ V the unique vertex in T′ fixed by G_{v}; and define a map f′ : V′ → V by reversing the roles of T and T′. For all v ∈ V, G_{v} fixes f′ ∘ f(v), so we must have f′ ∘ f(v) = v; similar reasoning applies to f ∘ f′, and we see that f and f′ are inverses. The maps f and f′ are adjacency preserving since two vertices are adjacent iff their stabilizers intersect in a subgroup isomorphic to ℤ^{2}. It is now straightforward to see that f defines a G-equivariant isomorphism T → T′.

Lemma 3.7 justifies use of the phrase “G ↷^{ρ} T is the Bass-Serre tree of the admissible group G.”

3.2. Vertex spaces, edge spaces, and geometric data for admissible actions

Definition 3.8. We say that G ↷^{ρ} X is an admissible action if G is an admissible group, X is a Hadamard space, and the action is discrete, cocompact, and isometric.

For the remainder of this section G ↷ X will be a fixed admissible action. In particular, all constants depend on G ↷ X (i.e. the group G, the Riemannian manifold X, and the action) in addition to other explicitly mentioned quantities. By Lemma 3.7 there is an essentially unique admissible graph of groups associated with G, and we will let G ↷ T be the corresponding Bass-Serre tree.

We refer the reader to section 2.3 for properties of Minsets that we use here. For each v ∈ V we let Y_{v} := Minset(Z(G_{v})) := ∩_{g∈Z(G_{v})}Minset(g) (this will be the Minset of a generator), and for every e ∈ E we let Y_{e} := Minset(G_{e}) := ∩_{g∈G_{e}}Minset(g).

Minset(α) is a convex subset of X, invariant under the centralizer of α, which is a metric product of ℝ with a Hadamard space. If α belongs to a group of isometries that acts cocompactly on X then the centralizer of α acts cocompactly on Minset(α) (see section 2.3). Thus Y_{v} is the product of ℝ with a Hadamard space Ȳ_{v}. Z(G_{v}) acts by translation on the ℝ factor and the induced action of H_{v} on Ȳ_{v} is discrete and cocompact. Y_{e} is the product of ℝ^{2} with a compact Hadamard space Ȳ_{e}, and G_{e} = ℤ^{2} acts by translations on the ℝ^{2} factor (section 2.3).

Note that the assignments v ↦ Y_{v} and e ↦ Y_{e} are G-equivariant with respect to the natural G actions. The minimal displacement of a generator of Z(G_{v}) is the same as that of a generator of Z(G_{g(v)}) = gZ(G_{v})g^{−1}. By the finiteness of G there is a number C such that for all v ∈ V the minimal displacement of a generator of Z(G_{v}) is less than C.

Definition 3.9. Let G ↷^{ρ} X be an admissible action, and let T be the Bass-Serre tree for G. For each v ∈ V we choose a generator ζ_{v} ∈ Z(G_{v}) in a G-equivariant way. We have an isometric splitting Y_{v} ≃ Ȳ_{v} × ℝ, which is preserved by G_{v}. The choice of generator ζ_{v} defines an orientation of the ℝ factor of Y_{v}. We have a map MLS_{v} : G_{v} → ℝ_{+} which assigns to each g ∈ G_{v} the minimum displacement of the induced isometry Ȳ_{v} → Ȳ_{v}. MLS_{v} descends to G_{v}/Z(G_{v}) ≃ H_{v} since Z(G_{v}) acts trivially on Ȳ_{v}. We define a homomorphism τ_{v} : G_{v} → ℝ by sending g ∈ G_{v} to the signed distance that g translates the ℝ factor of Y_{v} ≃ Ȳ_{v} × ℝ. The collections of functions MLS_{v} and τ_{v} constitute the geometric data of the action. Both MLS_{v} and τ_{v} descend to functions of the vertex groups of the graph of groups G defining G; we will sometimes find it more convenient to think of the geometric data in this way.

We remark that it follows from the discreteness of the action H_{v} ↷ Ȳ_{v} that g ∈ G_{v} then MLS_{v}(g) = 0 iff g projects to an element of finite order in H_{v}.

Lemma 3.10. The collections {Y_{v}}_{v∈V} and {Y_{e}}_{e∈E} are locally finite. More precisely, for every R there is an N so that if x ∈ X then there are at most N elements σ ∈ V ∪E so that Y_{σ} ∩ B(x, R) ≠ ∅.

Proof. Suppose v ∈ V and p ∈ Y_{v}. Then p has displacement < C under the generators of Z(G_{v}). Therefore if p ∈ B(x, R), then x has displacement < 2R + C under the generators of Z(G_{v}). But there are only finitely many g ∈ G with d(g·x, x) < 2R+C; since Z(G_{v_{1}}) ∩ Z(G_{v_{2}}) = {e} when v_{1} ≠ v_{2}, the local finiteness of {Y_{v}}_{v∈V} follows. Similar reasoning proves the local finiteness of {Y_{e}}_{e∈E}. The fact that N can be chosen independent of x follows from the cocompactness of the G action.

The lemma implies that for any D, the collection of D-tubular neighborhoods of the Y_{·}’s is locally finite. We also have the following consequences:

Lemma 3.11. For every D there are only finitely many pairs (σ_{1}, σ_{2}) ∈ (V ∪ E) × (V ∪ E) – modulo the diagonal action of G – with N_{D}(Y_{σ_{1}}) ∩ N_{D}(Y_{σ_{2}}) ≠ ∅.

Proof. By the finiteness of [V ∪ E]/G we need only show that for fixed σ there are only finitely many σ_{2} modulo G_{σ} such that N_{D}(Y_{σ}) ∩ N_{D}(Y_{σ_{2}}) ≠ ∅. This follows from Lemma 3.10, since G_{σ} acts cocompactly on N_{D}(Y_{σ}) and hence for some g ∈ G_{σ} N_{D}(Y_{σ}) ∩ (N_{D}(Y_{g(σ_{2})}) intersects a fixed ball.

Lemma 3.12. For σ_{1}, σ_{2} ∈ V ∪ E, G_{σ_{1}} ∩ G_{σ_{2}} acts cocompactly on the intersection N_{D}(Y_{σ_{1}})∩N_{D}(Y_{σ_{2}}). Thus, in particular, the diameter of N_{D}(Y_{σ_{1}})∩N_{D}(Y_{σ_{2}})/[G_{σ_{1}}∩G_{σ_{2}}] is uniformly bounded by a function of D.

Proof. This follows from the local finiteness of the family {Y_{σ}}_{σ∈V ∪E} and the discreteness of the cocompact action G ↷ X. Pick D > 0 and σ_{1}, σ_{2} ∈ V ∪ E. If x_{k} ∈ N_{D}(Y_{σ_{1}}) ∩ N_{D}(Y_{σ_{2}}), we may choose a sequence g_{k} ∈ G_{σ_{1}} such that g_{k}(x_{k}) → x_{∞} for some x_{∞} ∈ N_{D}(Y_{σ_{1}}). Then g_{k}(σ_{2}) lies in a finite subset of V ∪E (since g_{k}(N_{D}(Y_{σ_{2}})) intersects some ball B(x_{∞}, R) for all k) so after passing to a subsequence if necessary we may assume that g_{k}σ_{2} is constant. Then g_{1}^{−1}g_{k} ∈ G_{σ_{1}} ∩ G_{σ_{2}} and (g_{1}^{−1}g_{k})(x_{k}) → g_{1}^{−1}(x_{∞}). Thus G_{v_{1}} ∩ G_{v_{2}} acts cocompactly. The second statement now follows from Lemma 3.11.

Lemma 3.13. For every D there is a D′ (depending only on D) such that if σ ∈ V ∪E separates σ_{1} ∈ V ∪ E from σ_{2} ∈ V ∪ E, then N_{D}(Y_{σ_{1}}) ∩ N_{D}(Y_{σ_{2}}) ⊂ N_{D′}(Y_{σ}). In particular, if T_{1} and T_{2} are the closures of distinct connected components of T − σ then

[∪_{σ̂⊂T_{1}}N_{D}(Yσ̂)] ∩ [∪_{σ̂⊂T_{2}}N_{D}(Yσ̂)] ⊂ N_{D′}(Y_{σ}).

Proof. Pick D > 0. Suppose (σ_{1}, σ_{2}, σ) is a triple with σ_{i}, σ ∈ V ∪ E, σ separates σ_{1} from σ_{2} in T, and N_{D}(Y_{σ_{1}}) ∩ N_{D}(Y_{σ_{2}}) ≠ ∅. Then G_{σ_{1}} ∩ G_{σ_{2}} ⊂ G_{σ} and G_{σ_{1}} ∩ G_{σ_{2}} acts cocompactly on N_{D}(Y_{σ_{1}}) ∩ N_{D}(Y_{σ_{2}}) by Lemma 3.12; hence d(Y_{σ}, ·) is bounded on N_{D}(Y_{σ_{1}}) ∩ N_{D}(Y_{σ_{2}}). By Lemma 3.11 there are only finitely many such triples (σ_{1}, σ_{2}, σ) modulo G, so the lemma follows.

Definition 3.14. Since G acts cocompactly on X we can now fix a D so that ∪_{v∈V} N_{D}(Y_{v}) = ∪_{e∈E}N_{D}(Y_{e}) = X. We define X_{v} := N_{D}(Y_{v}) for all v ∈ V. Let D′ denote the constant in the previous lemma, D′′ = max(D, D′), and set X_{e} := N_{D′′}(Y_{e}) for all e ∈ E. We will refer to the X_{v}’s and X_{e}’s as vertex spaces and edge spaces respectively.

We note that Lemma 2.3 implies that for any σ ∈ V ∪ E we have ∂_{∞}X_{σ} = ∂_{∞}Y_{σ}. We summarize the properties of vertex and edge spaces:

Lemma 3.15. There is a constant C_{1} with the following property.

1. ∪_{v∈V} X_{v} = ∪_{e∈E}X_{e} = X.

2. If ê ∈ E and T_{1} and T_{2} are the distinct connected components of T − Int(ê), then [∪_{v∈T_{1}}X_{v}]−Xê and [∪_{v∈T_{2}}X_{v}]−Xê are disjoint closed and open subsets of X−Xê; and ∪_{e∈T_{1}}X_{e} − N_{C_{1}}(Xê) and ∪_{e∈T_{2}}X_{e} − N_{C_{1}}(Xê) are disjoint closed and open subsets of X − N_{C_{1}}(Xê).

3. If σ_{1}, σ_{2} ∈ V ∪ E and X_{σ_{1}} ∩ X_{σ_{2}} ≠ ∅ then d_{T} (σ_{1}, σ_{2}) < C_{1}.

Proof. 1 and 2 follow from the definition of vertex/edge spaces and Lemma 3.13. By Lemma 3.11 we can choose C_{1} so that 3 holds.

Corollary 3.16. For any v ∈ V, ∂_{T} X_{v} is isometric to the metric suspension of an uncountable discrete space, and for every e ∈ E, ∂_{T} X_{e} is isometric to a standard circle. Pick v_{1}, v_{2} ∈ V.

1. If d(v_{1}, v_{2}) > 2, then ∂_{T} X_{v_{1}} ∩ ∂_{T} X_{v_{2}} = ∅.

2. If d(v_{1}, v_{2}) = 2 and v is the vertex in between v_{1} and v_{2}, then ∂_{T} X_{v_{1}} ∩ ∂_{T} X_{v_{2}} = ∂_{T} γ where γ ⊂ Y_{v} is a geodesic of the form {p}×ℝ ⊂ Ȳ_{v} ×ℝ = Y_{v}; i.e. ∂_{T} X_{v_{1}} ∩∂_{T} X_{v_{2}} is the pair of suspension points of ∂_{T} X_{v}.

3. If d(v_{1}, v_{2}) = 1, then ∂_{T} X_{v_{1}} ∩ ∂_{T} X_{v_{2}} = ∂_{T} X_{e} ≃ S^{1}, where e := \overline{v_{1}v_{2}}.

Proof. Since Y_{v} ≃ Ȳ_{v} × ℝ, we have ∂_{T} X_{v} = ∂_{T} Y_{v} = Σ(∂_{T} Ȳ_{v}), and since Ȳ_{v} admits a discrete cocompact action by the non-elementary hyperbolic group H_{v} := G_{v}/Z(G_{v}), ∂_{T} Y_{v} is a discrete set with the cardinality of ℝ. For all e ∈ E, Y_{e} ≃ Ȳ_{e} × ℝ^{2} where Ȳ_{e} is compact, so ∂_{T} X_{e} = ∂_{T} Y_{e} ≃ ∂_{T} ℝ^{2}, and the latter is the standard circle.

Pick v_{1}, v_{2} ∈ V, and choose R large enough that Z := N_{R}(X_{v_{1}}) ∩ N_{R}(X_{v_{2}}) ≠ ∅. Then ∂_{T} X_{v_{1}} ∩ ∂_{T} X_{v_{2}} = ∂_{T} Z. The lemma now follows from Lemmas 3.4 and 3.12.

Lemma 3.17. There is a constant C_{2} with the following property. Suppose v, v′ ∈ V, e_{1}, …, e_{n} ∈ E are the consecutive edges of the segment \overline{vv′} ⊂ T, x ∈ X_{v} and y ∈ X_{v′}. Then for 1 ≤ i ≤ n we can find points z_{i} ∈ \overline{xy} such that

1. d(z_{i}, X_{e_{i}}) < C_{2}

2. For all 1 ≤ i ≤ j ≤ n we have d(z_{i}, x) ≤ d(z_{j}, x).

3. For every p ∈ X we have #{z_{i} ∈ B(p, 1)} < C_{2}.

Proof. Pick v, v′ ∈ V, x ∈ X_{v}, and y ∈ X_{v′}. Suppose ê ∈ E and let T_{1} and T_{2} be the two connected components of T − Int(ê). If \overline{xy} ∩ N_{C_{1}}(Xê) = ∅ (hence in particular \overline{xy} ∩ Xê = ∅) then by the first part of 2 of Lemma 3.15, \overline{xy} is contained in one of the two disjoint open sets (∪_{e⊂T_{i}}X_{e}) − N_{C_{1}}(Xê) for i = 1 or i = 2 . It follows that \overline{xy} ∩ N_{C_{1}}(X_{e_{i}}) is nonempty for every 1 ≤ i ≤ n. Let w_{i} ∈ \overline{xy} be the point in \overline{xy} ∩ N_{C_{1}}(X_{e_{i}}) closest to x. Let z_{1} = w_{1}, and let z_{i} be the element of \{w_{i}, …, w_{n}\} closest to x. So we have either z_{i} = w_{i}, or z_{i} = w_{i′} for some i′ > i. In the latter case \overline{xz_{i}} ⊂ ∪_{e⊂T′}X_{e} (hence in particular z_{i} = w_{i′} ∈ ∪_{e⊂T′}X_{e}) where T′ ⊂ T is the component of T − Int(e_{i}) containing e_{1}, so by Lemma 3.13 we have z_{i} ∈ N_{D′}(X_{e_{i}}) where D′ depends only on C_{1}. If p ∈ X and z_{i} ∈ B(p, 1), then X_{e_{i}} ∩ B(p, 1 + D′) ≠ ∅ and thus Y_{e_{i}} ∩ B(p, 1 + D′ + D′′) ≠ ∅ so by Lemma 3.10 we have #{z_{i} ∈ B(p, 1)} < N where N depends only on D′. Setting C_{2} := max{D′, N}, the lemma follows.

3.3. Itineraries

Our next objective is to associate an itinerary to any ray \overline{pξ} ⊂ X which is not contained in a finite tubular neighborhood of a single vertex space; the itinerary of \overline{pξ} is a ray in T which (roughly speaking) records the sequence of vertex spaces visited by \overline{pξ}.

Let ρ : X → V ⊂ T be a G-equivariant coarse Lipschitz map from the Hadamard space X to the vertex set of the tree T with the property that for every x ∈ X we have x ∈ X_{ρ(x)}. Such a ρ may be constructed as follows. Let Σ ⊂ X be a set theoretic cross-section for the free action G ↷ X; define ρ_{0} : Σ → T so that σ ∈ X_{ρ_{0}(σ)} for every σ ∈ Σ, and then extend ρ_{0} to an equivariant map X → T. Let L be such that N_{2D}(Y_{v_{1}}) ∩ N_{2D}(Y_{v_{2}}) ≠ ∅ implies d(v_{1}, v_{2}) < L, which exists by Corollary 3.11. In particular if d(x, y) < 2D then d(ρ(x), ρ(y)) ≤ L. In general, by dividing \overline{xy} into less than \frac{d(x,y)}{2D} + 1 segments of length less than 2D and adding the previous estimates we see that ρ will be coarse Lipschitz; i.e. d(ρ(x), ρ(y)) ≤ \frac{L}{2D} d(x, y) + L.

Lemma 3.18. If γ : [0, ∞) → X is a geodesic, then ρ ∘ γ : [0, ∞) → T has the bounded backtracking property^{7}.

Proof. Let e ∈ E be an edge in T, and let T_{1}, T_{2} ⊂ T be the connected components of T − Int(e). Suppose ρ ∘ γ(t_{1}) ∈ T_{1}, and ρ ∘ γ(t_{2}) ∈ T_{2} − N_{C_{1}}(e), where t_{2} > t_{1}. By Lemma 3.13 we have

[∪_{v∈T_{1}}X_{v}] ∩ [∪_{v∈T_{2}}X_{v}] ⊂ X_{e}.

Therefore there is a t_{3} ∈ [t_{1}, t_{2}) such that γ(t_{3}) ∈ X_{e}. Since d(ρ ∘ γ(t_{2}), e) ≥ C_{1}, the choice of C_{1} and lemma 3.15 implies that γ(t_{2}) ∉ X_{e}. Hence the convexity of X_{e} gives γ([t_{2}, ∞)) ⊂ X − X_{e}, which forces ρ ∘ γ([t_{2}, ∞)) ⊂ T_{2}. This property clearly implies uniformly bounded backtracking.

Lemma 3.19. If γ : [0, ∞) → X is a geodesic ray, then one of the following holds:

1. ρ ∘ γ : [0, ∞) → T is unbounded, and ρ ∘ γ([0, ∞)) lies in a uniform tubular neighborhood of a unique geodesic ray, τ, in T starting at ρ(γ(0)). The geodesic γ intersects X_{e} for all but finitely many edges e of τ.

7A map c : [0, ∞) → T has the bounded backtracking property if for every r ∈ (0, ∞) there is an r′ ∈ (0, ∞) such that if t_{1} < t_{2}, and d(c(t_{1}), c(t_{2})) > r′, then d(c(t), c(t_{1})) > r for every t > t_{2}.

2. ρ ∘ γ : [0, ∞) → T is bounded, and γ eventually lies in N_{D′}(Y_{v}) (where D′ comes from Definition 3.14) for some v ∈ V. In this case there is a subcomplex T_{γ} ⊂ T defined by the property that for each simplex σ in T, σ ∈ T_{γ} if and only if γ is asymptotic to X_{σ} . The possibilities for T_{γ} are: a single vertex v ∈ V, a single edge e ∈ E along with its vertices, or the closed star \overline{Star(v)} for some v ∈ V.

Proof. Pick v ∈ V. By the convexity of N_{D′}(Y_{v}), either γ is eventually contained in N_{D′}(Y_{v}), or γ is eventually contained in X − N_{D′}(Y_{v}). In the latter case ρ ∘ γ eventually remains in a unique component of T − v, by Lemma 3.13.

If for every v ∈ V the ray γ eventually lies in X − N_{D′}(Y_{v}), then clearly ρ ∘ γ is unbounded and hence it must lie within uniform distance of a ray in T by the bounded backtracking property. So we may assume that γ is eventually contained in N_{D′}(Y_{v}) for some v ∈ V. We note that if e ∈ T_{γ} then any vertex v′ of e must also be in T_{γ} since Y_{e} ⊂ Y_{v′}. Also if v, v′ ∈ V and d(v, v′) > 2 then part 1 of Lemma 3.4 along with Lemma 3.12 says that for any K, N_{K}X_{v} ∩ N_{K}X_{v′} is compact so γ ∉ N_{K}X_{v} ∩ N_{K}X_{v′} and hence at most one of v and v′ can be in T_{γ}.

If there are vertices v_{1} and v_{2} in T_{γ} with d(v_{1}, v_{2}) = 2 and v is the vertex between them then part 1 of Lemma 3.4 along with Lemma 3.12 says that Z(G_{v}) acts cocompactly on N_{K}X_{v_{1}} ∩ N_{K}X_{v_{2}} which contains γ for some K, and hence there is a K′ such that for all t > 0 there is a g_{t} ∈ Z(G_{v}) such that d(γ(t), g_{t}(γ(0)) < K′ and hence γ stays a distance at most K′ + d(γ(0), Y_{v}) from a geodesic in the ℝ direction of Y_{v} = Ȳ_{v} × ℝ (since Z(G_{v}) translates the ℝ direction). Thus for every e with v as a vertex we have γ is asymptotic to a geodesic in Y_{e} and hence e ∈ T_{γ}. Thus \overline{Star(v)} ⊂ T_{γ}. But since vertices in T_{γ} are at most distance 2 apart we see that \overline{Star(v)} = T_{γ}.

The only cases left for T_{γ} are the two mentioned and the case of two vertices a unit distance apart. But in the final case a similar argument shows that if e is the edge between them then γ stays a bounded distance from Y_{e} and hence e must also be in T_{γ}.

Definition 3.20. Let γ be a geodesic ray in X. If case 1 of Lemma 3.19 applies then we will say that γ has itinerary τ, and otherwise we say that the itinerary of γ is the subtree T_{γ} ⊂ T described in case 2 of the lemma. In either case we denote the itinerary of γ by I(γ).

One immediate consequence of the proof of Lemma 3.19 is

Corollary 3.21. If I(γ) = \overline{Star(v)} then γ is asymptotic to either the positive or the negative ℝ direction in the decomposition Y_{v} = Ȳ_{v} × ℝ.

Lemma 3.22. If \overline{p_{1}ξ}, \overline{p_{2}ξ} ⊂ X are asymptotic geodesic rays, then either both I(\overline{p_{1}ξ}) and I(\overline{p_{2}ξ}) are finite subtrees, in which case they agree, or both I(\overline{p_{1}ξ}) and I(\overline{p_{2}ξ}) are rays, in which case ∂_{∞}I(\overline{p_{1}ξ}) = ∂_{∞}I(\overline{p_{2}ξ}). In other words, I(\overline{p_{1}ξ}) is a ray in T if and only if I(\overline{p_{2}ξ}) is a ray in T asymptotic to I(\overline{p_{1}ξ}).

Proof. \overline{p_{1}ξ} lies in a tubular neighborhood of some Y_{v} if and only if \overline{p_{2}ξ} does, thus the case where I(\overline{p_{1}ξ}) (or I(\overline{p_{2}ξ})) is finite follows. Thus \overline{p_{1}ξ} has itinerary a ray τ_{1} if and only if \overline{p_{2}ξ} has itinerary τ_{2} for some ray τ_{2} ⊂ T. But the sets ρ(\overline{p_{1}ξ}) and ρ(\overline{p_{2}ξ}) are at finite Hausdorff distance from one another since ρ is coarse Lipschitz; hence the τ_{i} are asymptotic.

By the lemma we have a well-defined G-equivariant map from ∂_{∞}X to the union

∂_{∞}T ∪ (finite subsets of T)

which assigns to each ξ ∈ ∂_{∞}X either ∂_{∞}I(\overline{pξ}), p ∈ X if I(\overline{pξ}) is a ray or I(\overline{pξ}) otherwise; we will also denote this map by I. If η ∈ ∂_{∞}T, we use ∂_{∞}^{η}X to denote the corresponding subset: ∂_{∞}^{η}X := I^{−1}(η) ⊂ ∂_{∞}X. We will say that ∂_{∞}^{η}X is trivial if ∂_{∞}^{η}X is a point or nontrivial otherwise; (in the latter case we will see that ∂_{∞}^{η}X is homeomorphic to a closed interval and is in fact an interval in the Tits metric.).

In particular ∂_{∞}X = (∪_{v∈V} ∂_{∞}X_{v}) ∪ (∪_{η∈∂_{∞}T} ∂_{∞}^{η}X), where ∪_{v∈V} ∂_{∞}X_{v} is disjoint from ∪_{η∈∂_{∞}T} ∂_{∞}^{η}X.

The cone topology and Tits metric on ∂_{∞}X_{v} = ∂_{∞}Y_{v} = ∂_{∞}(Ȳ_{v} × ℝ) is described in sections 2.2 and 2.4. We see that in the cone topology ∂_{∞}X_{v} is just the suspension Σ(∂_{∞}H_{v}) and is independent of the metric on X. The Tits metric is just the metric suspension of the discrete metric.

We now study the dynamics of the action of G on ∂_{∞}X.

Lemma 3.23. 1. For every v ∈ V, the fixed point set of Z(G_{v}) in ∂_{∞}X is ∂_{∞}X_{v}; this set is homeomorphic to the suspension of ∂_{∞}H_{v} where H_{v} is the nonelementary hyperbolic group G_{v}/Z(G_{v}).

2. For every e ∈ E, F ix(G_{e}, ∂_{∞}X) = ∂_{∞}X_{e} which is homeomorphic to a circle.

Proof. Let ξ ∈ ∂_{∞}X be fixed by Z(G_{v}) and p ∈ Y_{v}. If I(\overline{pξ}) is a ray then by Lemma 3.22 ∂_{∞}I(\overline{pξ}) is fixed by Z(G_{v}). But this can not happen since by Lemma 3.5 Z(G_{v}) leaves no point in ∂_{∞}T fixed. So I(\overline{pξ}) is a finite subtree which by Lemma 3.22 is invariant under Z(G_{v}). Thus by Lemma 3.19 and Lemma 3.5 v ∈ I(\overline{pξ}). Thus \overline{pξ} ⊂ Y_{v} and hence ξ ∈ ∂_{∞}X_{v} = ∂_{∞}Y_{v}. On the other hand geodesics rays \overline{pξ} in Y_{v} are translated by a fixed amount by elements g ∈ G_{v}, so g(\overline{pξ}) is asymptotic to \overline{pξ} and so ξ is fixed by g. The rest of part 1 follows from sections 2.2 and 2.4 as above.

Let ξ be fixed by G_{e} and p ∈ Y_{e}. Again since G_{e} leaves no point in ∂_{∞}T fixed, by Lemma 3.5 I(\overline{pξ}) is a finite subtree that is invariant under G_{e} and hence by Lemma 3.19 and Lemma 3.5 must contain e. Hence we see \overline{pξ} ⊂ Y_{e} and ξ ∈ ∂_{∞}Y_{e} = ∂_{∞}X_{e}. Again, since G_{e} acts by translations on Y_{e}, we see that if ξ ∈ ∂_{∞}X_{e} = ∂_{∞}Y_{e} then it is left fixed by G_{e}. Since Y_{e} = ℝ^{2} × \overline{Y_{e}} where \overline{Y_{e}} is compact, ∂_{∞}Y_{e} = ∂_{∞}ℝ^{2} is homeomorphic to a circle.

4. Templates and the behavior of their geodesics

4.1. Templates

In this section we study “Templates”. These are piecewise Euclidean Hadamard spaces (which can be embedded in ℝ^{3}) which approximate certain subspaces of the spaces we are studying, and carry much of the information about the spaces at infinity.

A template is a Hadamard space T obtained from a disjoint collection of Euclidean planes {W }_{W ∈W all_{T}} (called walls) and directed Euclidean strips^{8} {S}_{S∈Strip_{T}} by isometric gluing^{9} subject to the following conditions:

1. The boundary geodesics of each strip S ∈ Strip_{T}, which we will refer to as singular geodesics, are glued isometrically to distinct walls in W all_{T}.

2. Each wall W ∈ W all_{T} is glued to at most two strips, and the gluing lines are not parallel.

3. T is connected. One can think of T as sitting in ℝ^{3} so that its walls are parallel planes and the strips meet the walls orthogonally. Two walls W_{1}, W_{2} ∈ W all_{T} are adjacent if there is a strip S ∈ Strip_{T} with S ∩ W_{i} ≠ ∅. The incidence graph Graph(T ) of T – the graph with vertex set W all_{T} and one edge for each pair of incident walls – is a graph isomorphic to a connected subcomplex of ℝ with the usual triangulation (where the vertices are the integers). A wall is an interior wall if it is incident to two strips, and a strip is an interior strip if it is incident to two interior walls; W all_{T}^{o} and Strip_{T}^{o} denote the interior walls and strips respectively. For every interior wall W ∈ W all_{T}^{o} we have a distinguished point o_{W} := W ∩ S_{1} ∩ S_{2}, where S_{i} ∈ Strip_{T}, i = 1, 2, are the strips incident to W. Let Strip_{T}^{+} be the collection of oriented interior strips; an orientation of a strip S ∈ Strip_{T} combines with the direction of S to give an orientation of the interval factor of S ≃ ℝ × I, and also an ordering of the two incident walls. We can define a function ϵ : Strip_{T}^{+} → ℝ as follows: if W_{−}, W_{+} are incident to S^{+} ∈ Strip_{T}^{+} and W_{−} < W_{+} with respect to the ordering defined by S^{+}, then ϵ(S^{+}) ∈ ℝ is defined to be the signed distance that o_{W_{+}} lies “above” o_{W_{−}} in the strip S^{+}. We also have a strip width function l : Strip_{T} → (0, ∞) and an angle function α : W all_{T}^{o} → (0, π) which give the angle between the oriented lines W ∩ S_{i} where S_{1}, S_{2} are incident to W.

We will sometimes enumerate the consecutive walls and strips of T so that W all_{T} = {W_{i}}_{a<i<b} and Strip_{T} = {S_{i}}_{a<i<b−1} where a ∈ {−∞, 0} and b ∈ ℕ ∪ {∞}. We then define L_{i}^{−} := W_{i} ∩ S_{i−1} for a + 1 < i < b and L_{i}^{+} := W_{i} ∩ S_{i} for a < i < b − 1.

An equivalence between two templates T_{1} and T_{2} is an isometry ψ : T_{1} → T_{2} which respects strip directions. Two templates are equivalent if and only if there is an incidence preserving bijection W all_{T_{1}} ∪ Strip_{T_{1}} → W all_{T_{2}} ∪ Strip_{T_{2}} which respects the functions l, ϵ, and α. We will call a template uniform if there is a \frac{π}{2} ≥ β > 0 so that the angle function α : W all_{T}^{o} → (0, π) satisfies π − β ≥ α ≥ β, and if the strip widths are bounded away from zero. We are mostly interested in uniform templates. A template T is full if Graph(T ) ≃ ℝ, half if Graph(T ) ≃ ℝ_{+}, and finite if |W all_{T} | < ∞.

If W ∈ W all_{T}^{o} and S_{1}, S_{2} are the incident strips, then the oriented lines W ∩ S_{1} and W ∩ S_{2} divide the plane W into four sectors which we call Quarter Planes and which we label as Q_{I}, Q_{II}, Q_{III}, and Q_{IV} as usual. If we are given a choice Q_{W} of quarter planes in W for each W ∈ W all_{T} then there is an isometric immersion D : [∪_{W ∈W all_{T}} Q_{W}] ∪ [∪_{S∈Strip_{T}} S] → ℝ^{2} (the development) which takes any geodesic ray γ ⊂ T such that γ ∩ W ⊂ Q_{W}, to a Euclidean ray (see Figure 3 in section 7.2 for an example of the developement of a special kind of template).

8A direction for a strip S is an orientation for its ℝ-factor S ≃ ℝ × I.

9In general one may also have to complete the resulting quotient space to get a Hadamard space.

When T is a half template we will be primarily interested in geodesic rays γ ⊂ T that start at a given base point and intersect all but finitely many walls of T . From the separation properties of walls it is clear that such a ray intersects the walls W ∈ W all_{T} in order. We let ∂_{∞}^{∞}T (resp. ∂_{T}^{∞}T ) denote the corresponding subset of ∂_{∞}T (resp. ∂_{T} T ). In section 7.1 we will show that ∂_{T}^{∞}T is isometric to either a point, in which case T is called trivial, or an interval of length < π. Remark 4.1. One can show directly that any two half templates such that corresponding angles agree, and both corresponding strip widths and displacements differ by a bounded amount will have ∂_{∞}^{∞}’s with the same Tits length. We will only need a weaker version (that will follow from Theorem 5.1) in this paper so we will not digress to prove it here.

4.2. Templates associated with itineraries in T

We now return to the setting of our paper: G is an admissible group with a discrete cocompact isometric action on a Hadamard space X. We now want to associate a template with each geodesic segment/ray in T; these templates capture the asymptotic geometry of geodesic segments/rays in X which pass near the corresponding edge spaces.

We first choose, in a G-equivariant way, a plane F_{e} ⊂ Y_{e} for each edge e ∈ E. Then for every pair of adjacent edges e_{1}, e_{2} we choose, again equivariantly, a minimal geodesic from F_{e_{1}} to F_{e_{2}}; by the convexity of Y_{v} = Ȳ_{v} × ℝ, v := e_{1} ∩ e_{2}, this geodesic determines a Euclidean strip^{10} S_{e_{1},e_{2}} := γ_{e_{1},e_{2}} × ℝ for some geodesic segment γ_{e_{1},e_{2}} ⊂ Ȳ_{v}; note that S_{e_{1},e_{2}} ∩ F_{e_{i}} is an axis of Z(G_{v}). Hence if e, e_{1}, e_{2} ∈ E, e_{i} ∩ e = v_{i} ∈ V are distinct vertices, then the angle between the geodesics S_{e_{1},e} ∩ F_{e} and S_{e_{2},e} ∩ F_{e} is bounded away from zero (since only finitely many angles show up). We also note that Definition 3.14 tells us that d_{H}(F_{e}, X_{e}) is bounded by D′′ + Diam(Ȳ_{e}) which, since there are only finitely many e up to the action of G, is uniformly bounded.

The significance of the strips S_{e_{1},e_{2}} can be seen in the next two lemmas.

Lemma 4.2. There is a constant C_{3} so that if e_{1} = \overline{vv_{1}} and e_{2} = \overline{vv_{2}} are adjacent edges, then X_{e_{1}} ∪ S_{e_{1},e_{2}} ∪ X_{e_{2}} and X_{v_{1}} ∪ S_{e_{1},e_{2}} ∪ X_{v_{2}} are C_{3}-quasi-convex.

Proof. Since the Hausdorff distance d_{H}(Xê, Fê) is uniformly bounded for ê ∈ E, it suffices to show that there is a constant C so that the unions F_{e_{1}} ∪ S_{e_{1},e_{2}} ∪ F_{e_{2}} are C-quasi-convex for all pairs of adjacent edges. But if e_{1}, e_{2} ∈ E are adjacent then F_{e_{1}} ∪S_{e_{1},e_{2}} ∪F_{e_{2}} ⊂ Y_{v} ≃ Ȳ_{v} ×ℝ, and we are reduced to showing that F̄_{e_{1}} ∪γ_{e_{1},e_{2}} ∪F̄_{e_{2}} ⊂ Ȳ_{v} is uniformly quasi-convex, where F̄_{e_{i}} is the image of F_{e_{i}} under the projection Y_{v} ≃ Ȳ_{v} × ℝ → Ȳ_{v}. This follows from Lemma 2.9. This gives the first statement.

The interesting part of the second statement is when we consider \overline{xy} when x ∈ X_{v_{1}} and y ∈ X_{v_{2}}. In this case Lemma 3.17 gives us z_{1} ∈ N_{C_{2}}X_{e_{1}} and z_{2} ∈ N_{C_{2}}X_{e_{2}} on \overline{xy}. Now the first statement along with the convexity of N_{C_{2}+D′′}(X_{v_{i}}) (note z_{i} ∈ N_{C_{2}}X_{e_{i}} ⊂ N_{C_{2}+D′′}X_{v_{i}}) yields the second statement.

10S_{e_{1},e_{2}} may have width zero.

Lemma 4.3. There is a constant C_{4} so that if e_{1}, …, e_{n} ∈ E is a geodesic edge path in T with initial vertex v_{1} and terminal vertex v_{n}, then

Z := X_{e_{1}} ∪ S_{e_{1},e_{2}} ∪ X_{e_{2}} ∪ … ∪ X_{e_{n−1}} ∪ S_{e_{n−1},e_{n}} ∪ X_{e_{n}}

and

Z′ := X_{v_{1}} ∪ S_{e_{1},e_{2}} ∪ X_{e_{2}} ∪ … ∪ X_{e_{n−1}} ∪ S_{e_{n−1},e_{n}} ∪ X_{v_{n}}

are C_{4}-quasi-convex.

Proof. Pick x, y ∈ Z. We may assume without loss of generality that x ∈ X_{e_{1}}∪S_{e_{1},e_{2}}∪ X_{e_{2}} ⊂ X_{v} where v = e_{1}∩e_{2}, and y ∈ X_{e_{n−1}} ∪S_{e_{n−1},e_{n}} ∪X_{e_{n}} ⊂ X_{v′} where v′ = e_{n−1}∩e_{n}. Applying Lemma 3.17 we get points z_{i} ∈ \overline{xy} ∩ N_{C_{2}}(X_{e_{i}}) for 2 ≤ i ≤ n − 1, with d(z_{i}, x) ≤ d(z_{j}, x) when i ≤ j. If C_{4} := C_{2} + C_{3}, then by Lemma 4.2 we have \overline{xz_{2}} ⊂ N_{C_{4}}(X_{e_{1}} ∪ S_{e_{1},e_{2}} ∪ X_{e_{2}}), \overline{z_{i}z_{i+1}} ⊂ N_{C_{4}}(X_{e_{i}} ∪ S_{e_{i},e_{i+1}} ∪ X_{e_{i+1}}) for i = 2, … n − 1, and \overline{z_{n−1}y} ⊂ N_{C_{4}}(X_{e_{n−1}} ∪ S_{e_{n−1},e_{n}} ∪ X_{e_{n}}).

We omit the proof that Z′ is quasi-convex, as it is similar.

Lemma 4.3 suggests that we will understand the geodesic geometry of X if the geometry of the sets Z (as in the lemma) can be easily modeled. To this end, we “approximate” Z with a template.

Definition 4.4. Suppose γ ⊂ T is a geodesic segment or ray. Let T be a template with walls {W }_{W ∈W all_{T}} and strips {S}_{S∈Strip_{T}}, let f : W all_{T} → E be an adjacency preserving bijection between the walls of T and the edges of γ, and let ϕ : T → X be a (not necessarily continuous) map. Then the triple (T , f, ϕ) is a K-template for γ if for all W ∈ W all_{T} we have ϕ(W) ⊂ N_{K}(X_{f(W)}) and X_{f(W)} ⊂ N_{K}(ϕ(W)) and the following conditions are met for every S ∈ Strip_{T}.

1. W idth(S) ≥ 1.

2. If S is incident to W_{1}, W_{2} ∈ W all_{T} and x, y ∈ W_{1} ∪ S ∪ W_{2} then we have |d_{X}(ϕ(x), ϕ(y)) − d_{T} (x, y)| < K and if γ_{1} : [0, 1] → \overline{xy} and γ_{2} : [0, 1] → \overline{ϕ(x)ϕ(y)} are constant speed parameterizations, then d(ϕ ∘ γ_{1}(t), γ_{2}(t)) < K for all t ∈ [0, 1].

3. If W_{1}, W_{2} ∈ W all_{T} are adjacent to S then the Hausdorff distance d_{H}((X_{f(W_{1})} ∪ S_{f(W_{1}),f(W_{2})} ∪ X_{f(W_{2})}), ϕ(W_{1} ∪ S ∪ W_{2})) < K.

Often when the value of K is not relevant we will refer to the triple (T , f, ϕ) as a template for γ, by which we mean a K-template for some K.

Let T′ be another template such that the angles on corresponding walls agree with those of T and such that the other data (strip widths and displacements) differ from T by a bounded amount. Then there is a natural (discontinuous) map F : T′ → T which is an isometry on each wall and simply stretches the width of the strips. It is easy to check that using ϕ′ = ϕ ∘ F that we get a K′ template (T′, f, ϕ′) for γ (see step 3 of the proof of Lemma 7.14).

For a suitably large K we describe a construction that gives a K-template for any geodesic segment or geodesic ray γ ⊂ T . We will refer to these K-templates as standard K-templates. We begin with a disjoint collection of walls W_{e} and an isometry ϕ_{e} : W_{e} → F_{e} for each edge e ⊂ γ. For every pair e, e′ of adjacent edges of γ, we let Ŝ_{e,e′} be a strip which is isometric to S_{e,e′} ⊂ X if W idth(S_{e,e′}) ≥ 1, and isometric to ℝ × [0, 1] otherwise; we let ϕ_{e,e′} : Ŝ_{e,e′} → S_{e,e′} be an affine map which respects product structure (ϕ_{e,e′} is an isometry if W idth(S_{e,e′}) ≥ 1 and compresses the interval otherwise). We construct T by gluing the strips and walls so that the maps ϕ_{e} and ϕ_{e,e′} descend to continuous maps on the quotient.

The above construction yields

Lemma 4.5. There is a constant K = K(X) such that for every geodesic segment or ray, γ ⊂ T, there is a K-template for γ.

There is a β = β(X) > 0 such that for any K-template the angle function α : W all_{T}^{o} → (0, π) satisfies 0 < β ≤ α ≤ π − β < π.

Proof. We check that each condition of Definition 4.4 holds for the standard template described above, for sufficiently large K.

First, since d_{H}(F_{e}, X_{e}) is uniformly bounded and F_{f(W)} = ϕ(W) for every W ∈ W all_{T}, we have ϕ(W) ⊂ N_{K}(X_{f(W)}) and X_{f(W)} ⊂ N_{K}(ϕ(W)) for all W ∈ W all_{T} for large enough K.

Conditions 1 and 3 follow immediately from the description of standard templates. We now verify condition 2. Pick adjacent walls W, W′ ∈ W all_{T} and set e := f(W),

e′ := f(W′), and v := e∩e′. Recall that F_{e}∪S_{e,e′} ∪F_{e′} ⊂ Y_{v} and Y_{v} splits isometrically as Y_{v} = Ȳ_{v} × ℝ where Ȳ_{v} is Gromov hyperbolic. Furthermore, ϕ induces a map W_{e} ∪ Ŝ_{e,e′} ∪ W_{e′} → F_{e} ∪ S_{e,e′} ∪ F_{e′} which is compatible with the product structure. Hence condition 2 follows from part 3 of Lemma 2.9 (and triangle inequalities) when d(F_{e}, F_{e′}) ≥ 4δ (where δ is the maximum of the hyperbolicity constants of the Ȳ_{v}’s); modulo G there are only finitely many cases when d(F_{e}, F_{e′}) < 4δ (Lemma 3.11), and each of these is also settled by part 3 of Lemma 2.9.

For any K template (T , f, ϕ) for a γ containing an interior edge e = \overline{v′v} we claim that the wall W with f(W) = e will have the same angle, up to taking supplements (i.e. α might be replaced by π − α), as the angle α between the ℝ factors of Y_{v′} = Ȳ_{v′} × ℝ and Y_{v} = Ȳ_{v} × ℝ in Y_{e} = Y_{v} ∩ Y_{v′}. The fact that these angles are positive and the finiteness of edges modulo G will yield the result. We note that α is the Tits angle between the ℝ factors.

To see this we first note that Property 2 of Definition 4.4 says that the angle for W, i.e. the angle between the gluing lines L′ and L, is the same as the comparison angle lim_{t→∞} ˜∠_{ϕ(o)}ϕ(L′(t)), ϕ(L(t)). Also Property 2 of Definition 4.4 says that there are geodesic rays σ = lim_{i→∞} \overline{pϕ(L(t_{i}))} where t_{i} → ∞ and p ∈ Y_{e}. Since ϕ(W) ⊂ N_{K}X_{e} we see that ϕ(L) ⊂ N_{K}X_{e} and hence e ∈ I(σ). Now if we let e′ be the other edge incident to v in γ then the intersection of the wall f^{−1}(e′) with the strip between W and f^{−1}(e′), is a line parallel to L and hence, again by 2 of Definition 4.4, ϕ(L) stays in a uniform neighborhood of X_{e′} so e′ ∈ I(σ). but e, e′ ∈ I(σ) implies by Lemma 3.19 that I(σ) = \overline{Star(v)} and hence by Corollary 3.21 any such σ is asymptotic to the ℝ factor of Y_{v} = Ȳ_{v} × ℝ. Since p ∈ Y_{e} ⊂ Y_{v}, σ is a half line of such an ℝ and we assume without loss of generality that it points in the positive direction. A similar argument works for ϕ(L′). Thus ∠(L′, L) = lim_{t→∞} ˜∠_{ϕ(o)}ϕ(L′(t)), ϕ(L(t)) ≤ α. Also the same arguments applied to −L′ and L yield ∠(−L′, L) =≤ π − α. Thus we get equality and the result.

The next proposition and Proposition 4.8 are technical results that compare template geometry with ambient geometry.

Proposition 4.6. Suppose K > 0. There is a constant C_{5} depending only on K and the geometry of X with the following property. Suppose γ ⊂ T is a geodesic segment or geodesic ray in T with i^{th} edge e_{i}, and set Z := [∪_{e⊂γ}X_{e}]∪[∪_{e,e′⊂γ}S_{e,e′}]. If (T , f, ϕ) is a K-template for γ, and x, y ∈ Z, then there is a continuous map α : \overline{xy} → T so that

1. d(ϕ ∘ α, id|\overline{xy}) < C_{5}

2. For all p, q ∈ \overline{xy} we have

d_{X}(p, q) − kC_{5} ≤ d_{T} (α(p), α(q)) ≤ length(α|\overline{pq}) ≤ d_{X}(p, q) + kC_{5}  (4.7)

where the segment \overline{α(p)α(q)} ⊂ T intersects at most k − 1 strips and walls in T . In particular there are constants (L, A) depending only on K and X so that ϕ is an (L, A) quasi-isometric embedding for every K-template (T , f, ϕ).

Proof. By the standard properties of Hadamard spaces we may reduce to the case that x ∈ Y_{e_{x}} ∪S_{e_{x},e′_{x}} ⊂ X_{v_{x}} (v_{x} = e_{x}∩e′_{x}) and y ∈ S_{e_{y},e′_{y}} ∪Y_{e′_{y}} ⊂ X_{v_{y}} since the original x and y are within a bounded distance of such. Let W_{i} := f^{−1}(e_{i}) ∈ W all_{T} for e_{i} 1 ≤ i ≤ n the edges between v_{x} and v_{y}. We may apply Lemma 3.17 to the pair x, y obtaining points z_{i} ∈ \overline{xy}. We let z_{0} = x and z_{n+1} = y. After making a small perturbation of the z_{i}’s if necessary, we may assume that they satisfy d_{X}(z_{i}, x) < d(z_{j}, x) when i < j. For 1 ≤ i ≤ n pick w_{i} ∈ T with w_{i} ∈ W_{i} ⊂ T with d(z_{i}, ϕ(w_{i})) ≤ 1 + inf{d(z_{i}, ϕ(w)) | w ∈ W_{i}} ≤ C_{2} + 1. By Definition 4.4 part 3 we can also choose w_{0} ∈ W_{0} ∪ S ∪ W_{1} and such that d_{X}(x, ϕ(w_{0})) ≤ K and similarly choose w_{n+1}. Now define α by the condition that α(z_{i}) = w_{i}, and α is a constant speed geodesic on the segment \overline{z_{i}z_{i+1}}.

Proof of 1. Apply Definition 4.4 to see that the constant speed parameterization [0, 1] → \overline{ϕ(w_{i})ϕ(w_{i+1})} is at uniformly bounded distance from the composition of the constant speed parameterization [0, 1] → \overline{w_{i}w_{i+1}} ⊂ T with ϕ : T → X. Since d(ϕ(w_{i}), z_{i}) is uniformly bounded, we know that the constant speed parameterizations [0, 1] → \overline{ϕ(w_{i})ϕ(w_{i+1})} and [0, 1] → \overline{z_{i}z_{i+1}} are also at uniformly bounded distance from one another, so there is a constant c_{1} depending on K so that d(ϕ ∘ α, id|\overline{xy}) < c_{1}.

Proof of 2. Assume p ∈ \overline{z_{j−1}z_{j}} − z_{j−1} and q ∈ \overline{z_{j′}z_{j′+1}} − z_{j′+1} for j ≤ j′. By Definition 4.4 we have, for c_{2} = 2c_{1} + K

|length(α|\overline{pz_{j}}) − d_{X}(p, z_{j})| < c_{2}
|length(α|\overline{z_{i}z_{i+1}}) − d_{X}(z_{i}, z_{i+1})| < c_{2} for every i = 1, … n − 1
|length(α|\overline{z_{j′}q}) − d_{X}(z_{j′}, q)| < c_{2}.

Hence there is a c_{3} = c_{3}(K) so that

length(α\overline{pq}) = length(α|\overline{pz_{j}}) + … + length(α|\overline{z_{j′}q})
≤ d_{X}(p, z_{j}) + … + d_{X}(z_{j′}, q) + (j′ − j + 2)c_{2}
≤ d_{X}(p, q) + kc_{2}.

To prove the remaining inequality of (4.7) we break up the T -geodesic \overline{α(p)α(q)} into at most k subsegments \overline{u_{i}u_{i+1}} so that each subsegment lies in W_{i} ∪ S_{i} ∪ W_{i+1} for some i. Then by definition 4.4 we have |d_{X}(ϕ(u_{i}), ϕ(u_{i+1})) − d_{T} (u_{i}, u_{i+1})| < K so

d_{X}(p, q) ≤ 2c_{1} + d_{X}(ϕ ∘ α(p), ϕ ∘ α(q))
≤ 2c_{1} + ∑ d_{X}(ϕ(u_{i}), ϕ(u_{i+1}))
≤ 2c_{1} + kc_{2} + d_{T} (α(p), α(q))
≤ d_{T} (α(p), α(q)) + kc_{3}.

where c_{3} := 2c_{1} + c_{2}.

To see the quasi-isometry property of ϕ, let x′, y′ ∈ T , x = ϕ(x′), y = ϕ(y′), and let α be the map defined above where we choose w_{0} = x and w_{1} = y (i.e α(x) = x′ and α(y) = y′). Now 4.7 applied to p = x and q = y along with k ≤ d_{T} (x′, y′) + 1 (since strips have width at least 1) and k ≤ const_{1}d_{X}(p, q)+const_{2} (as in the proof of the coarse lipschitz property of ρ - see section 3.3) yields the quasi-isometry property of ϕ. This completes the proof of Proposition 4.6.

Proposition 4.8. Pick K > 0. There is a constant C_{6} = C_{6}(K, X) so that the following holds. If (T , f, ϕ) is a K-template for γ ⊂ T, and x, y ∈ T , then there is a continuous map α : \overline{xy} → X where

1. d(α, ϕ|\overline{xy}) < C_{6}.

2. For all p, q ∈ \overline{xy} we have

d_{T} (p, q) − kC_{6} ≤ d_{X}(α(p), α(q)) ≤ length(α|\overline{pq}) ≤ d_{T} (p, q) + kC_{6}  (4.9)

where the segment \overline{pq} ⊂ T intersects at most (k − 1) strips and walls in T .

Proof. This is similar to the proof of Proposition 4.6, so we omit it.

Corollary 4.10. If (T , f, ϕ) is a K-template and C_{6} = C_{6}(K) is the constant from Proposition 4.8, then for any x, y ∈ T we have

d_{T} (x, y) − kC_{6} ≤ d_{X}(ϕ(x), ϕ(y)) ≤ d_{T} (x, y) + kC_{6}

where \overline{xy} ⊂ T meets at most k − 1 strips and walls.

5. Shadowing

In this section we show that geodesic segments in a K-template are sublinearly shadowed by ambient geodesic segments, and vice-versa.

Theorem 5.1 (Shadowing). There is a function θ : ℝ_{+} → ℝ_{+} depending on K and the geometry of X (sometimes denoted θ_{(X,K)}) with lim_{R→∞} θ(R) = 0 so that if (T , f, ϕ) is a K-template for a geodesic segment/ray γ ⊂ T, then the following hold.

1. If x, y ∈ T , z ∈ \overline{xy} and R := d(z, x), then d(ϕ(z), \overline{ϕ(x)ϕ(y)}) ≤ (1 + R)θ(R).

2. If x, y ∈ T , z ∈ \overline{ϕ(x)ϕ(y)} and R := d(z, ϕ(x)) then d(z, ϕ(\overline{xy})) ≤ (1 + R)θ(R).

3. Let T̄ := T ∪ ∂_{∞}T and X̄ := X ∪ ∂_{∞}X be the usual compactifications. Then there is a unique topological embedding ∂_{∞}ϕ : ∂_{∞}T → ∂_{∞}X so that

ϕ̄:= ϕ ∪ ∂_{∞}ϕ : T̄ → X̄

is continuous at every ξ ∈ ∂_{∞}T ⊂ T̄.

4. The image of ∂_{∞}ϕ is

[∪_{W ∈W all_{T}} (∂_{∞}X_{f(W)})] ∪ [∂_{∞}^{∂_{∞}γ}X],

and when γ is a ray with ∂_{∞}γ = η then ∂_{∞}ϕ(∂_{∞}^{∞}T ) = ∂_{∞}^{η}X (see section 3.3 for the definition of ∂_{∞}^{η}X).

5. ∂_{∞}ϕ|_{∂_{∞}^{∞}T} : ∂_{∞}^{∞}T → ∂_{∞}^{η}X is an isometric embedding with respect to the Tits metric.

The proof of the theorem breaks up into two pieces. We first show in Proposition 5.6 that a geodesic segment (in a template or in X) running through a sequence of consecutive walls has to be “close” to any point p which lies close to sufficiently many walls in the sequence. We then show in Theorem 5.7 that a segment in a template (resp. in X) which doesn’t meet too many walls (i.e. encounters at most Const log R walls in the segment \overline{px}, d(p, x) = R) is well shadowed by a geodesic segment in X (resp. in the template). These two arguments are combined in section 5.3 to prove Theorem 5.1.

5.1. Paths in a template which are close to a cluster of walls

We begin with a result about templates. It estimates the excess length of a path η which connects two walls W, W′ while remaining outside a ball which intersects W, W′, and all walls between them.

Proposition 5.2. Let T be a template with angle function α : W all_{T}^{o} → (0, π) satisfying 0 < β ≤ α ≤ π − β < π. Then there are positive constants N_{1} = N_{1}(β), (N_{1} ≈ \frac{Const}{β}), C_{1} = C_{1}(β), and C_{2} = C_{2}(β) with the following property. Let W_{n_{0}}, …, W_{n_{1}} ∈ W all_{T} be a sequence of consecutive walls, and suppose W_{i}∩B(p, R) ≠ ∅ for some p ∈ T , R > 0 and every n_{0} ≤ i ≤ n_{1}. Then for any R′ ≥ N_{1}R and any path c : [0, 1] → T − B(p, R′) with c(0) ∈ W_{n_{0}} and c(1) ∈ W_{n_{1}} we have

length(c) ≥ d_{T} (c(0), c(1)) + C_{1}(n_{1} − n_{0} − C_{2})R′.

Proof. Let S_{i} be the strip incident to W_{i} and W_{i+1} for i = n_{0}, …, n_{1} − 1, set L_{i}^{−} := S_{i−1} ∩ W_{i} for i = n_{0} + 1, … n_{1}, and set L_{i}^{+} := S_{i} ∩ W_{i} for i = n_{0}, …, n_{1} − 1.

We prove the proposition with the help of some lemmas.

Lemma 5.3. There is a constant c_{1} ≈ \frac{Const}{β} so that if p ∈ T and d(p, W_{j}) < R for j = i ± 1 then d(p, o_{i}) < c_{1}R.

Proof. By joining \overline{x_{i−1}p} to \overline{px_{i+1}} for appropriate choices of x_{i−1} ∈ W_{i−1} and x_{i+1} ∈ W_{i+1} we get a path γ : [0, 1] → T of length at most 2R joining W_{i−1} to W_{i+1}. Therefore there is a segment [a, b] ⊂ [0, 1] with γ(a) ∈ L_{i}^{−} and γ(b) ∈ L_{i}^{+}. So

d(p, o_{i}) ≤ R + min(d(γ(a), o_{i}), d(γ(b), o_{i}))

≤ R + \frac{d(γ(a), γ(b))}{2 sin(\frac{β}{2} )} ≤ c_{1}R.

We now define N_{1} := max(2c_{1}, [\frac{π}{β}] + 2).

Consider a path η : [0, 1] → T − B(p, R′) where R′ > N_{1}R. The ball B(p, R′) ⊂ T is convex, so clearly T − B(p, R′) is complete and locally compact with respect to the induced path metric. Therefore we may assume that c is a constant speed minimizing path from c(0) to c(1) in T − B(p, R′). Since \overline{B(p, R′)} is a convex subset of the Hadamard space T , the nearest point projection T → \overline{B(p, R′)} is distance non increasing; it follows that the set c^{−1}(\overline{B(p, R′)}) is either empty or a closed subinterval [a, b] ⊂ [0, 1]. c|_{[0,a]} and c|_{[b,1]} are constant speed geodesic segments in the Hadamard space T , and since R′ > c_{1}R these segments lie in T − {o_{i}}_{n_{0}<i<n_{1}} by Lemma 5.3.

Lemma 5.4.

c([0, a]) ⊂ [∪_{i=n_{0}}^{n_{0}+N_{1}}W_{i}] ∪ [∪_{i=n_{0}}^{n_{0}+N_{1}−1}S_{i}]

and

c([b, 1]) ⊂ [∪_{i=n_{1}−N_{1}}^{n_{1}}W_{i}] ∪ [∪_{i=n_{1}−N_{1}}^{n_{1}−1}S_{i}] .

Proof. We prove the first assertion; the proof of the second is similar. If the lemma were false, we would have c(t) ∈ S_{n_{0}+N_{1}} for some t ∈ [0, a]. Therefore c([0, t]) must cross every strip S_{i} for n_{0} ≤ i < n_{0} + N_{1}, and for every n_{0} < i ≤ n_{0} + N_{1} it must enter W_{i} through L_{i}^{−} and exit through L_{i}^{+}. Since c([0, a]) is disjoint from {o_{i}}_{n_{0}<i<n_{1}} there is a flat convex strip Y ⊂ T containing c([0, a]) in its interior. Using Y we can define co-orientations for the segments c([0, a]) ∩ S_{i} and c([0, a]) ∩ W_{i} for n_{0} ≤ i ≤ n_{0} + N_{1}. If two of the origins o_{i} ∈ W_{i} for n_{0} < i ≤ n_{0} + N_{1} lie on opposite sides of the corresponding segments c([0, a]) ∩ W_{i} with respect to the coorientations then the geodesic between them (of length less than 2c_{1}R by lemma 5.3) will intersect c([0, a]) and hence d(o_{i}, c([0, a])) < c_{1}R for some n_{0} < i ≤ n_{0} + N_{1}, and thus d(p, c([0, a]) < 2c_{1}R. But this cannot happen since d(p, c([0, a]) ≥ R′ > 2c_{1}R. Thus all the origins o_{i} ∈ W_{i} for n_{0} < i ≤ n_{0} + N_{1} lie on the same side of the corresponding segments c([0, a]) ∩ W_{i} with respect to the co-orientations. It follows that the angle between c([0, a]) and L_{i}^{−} increases by at least β each time c([0, a]) passes through a wall. Hence (N_{1} − 1)β < π, contradicting the definition of N_{1}.

Proof of Proposition 5.2 concluded. Let [a′, b′] ⊂ [a, b] ⊂ [0, 1] be the inverse image of

[∪_{i=n_{0}+N_{1}+1}^{n_{1}−N_{1}−1}W_{i}] ∪ [∪_{n_{0}+N_{1}}^{n_{1}−N_{1}−1}S_{i}]

under c. We know that c([a′, b′]) remains in the sphere S(p, R′) while it passes through all the walls W_{i} for n_{0} + N_{1} < i < n_{1} − N_{1}. So for every n_{0} + N_{1} < i < n_{1} − N_{1}, c([a′, b′]) joins L_{i}^{−} to L_{i}^{+} outside B(p, R′) ⊃ B(o_{i}, R′ −c_{1}R). Hence length(c([a′, b′]) ≥ β(R′−c_{1}R)(n_{1}−n_{0}−(2N_{1}+2)) ≥ \frac{β}{2} R′(n_{1}−n_{0}−(2N_{1}+2)) while d_{T} (c(a′), c(b′)) ≤ 2R′ so length(c[a′, b′]) ≥ d_{T} (c(a′), c(b′)) + \frac{β}{2} (n_{1} − n_{0} − (2N_{1} + 2) − \frac{4}{β})R′ and hence

length(c) ≥ d_{T} (c(0), c(1)) + C_{1}(n_{1} − n_{0} − C_{2})R′

where C_{1}, C_{2} depend only on β.

Corollary 5.5. Let T , N_{1}, C_{1}, C_{2}, W_{n_{0}}, …, W_{n_{1}}, p, R be as in Proposition 5.2. If n_{1} − n_{0} > C_{2}, then any geodesic segment from W_{n_{0}} to W_{n_{1}} must pass through B(p, N_{1}R).

The result corresponding to Corollary 5.5 in the space X is:

Proposition 5.6. There are constants N_{2} = N_{2}(X), R_{0} = R_{0}(X) with the following property. If n ≥ N_{2}, e_{1}, … e_{n} ∈ E are the consecutive edges of a geodesic segment, γ, in the tree T, p ∈ X, R ≥ R_{0}, and X_{e_{i}} ∩ B(p, R) ≠ ∅ for 1 ≤ i ≤ n; then for any C ≥ 0, and any segment \overline{xy} ⊂ X with \overline{xy} ∩ N_{C}(X_{e_{i}}) ≠ ∅ for i = 1 and i = n, we have \overline{xy} ∩ B(p, N_{2}R + 2C) ≠ ∅.

Proof. Let e_{1}, …, e_{n}, p, X_{e_{i}} be as in the statement of the proposition. If \overline{xy} ∩ N_{C}(X_{e_{i}}) ≠ ∅ for i = 1 and i = n, then we have x_{0} ∈ X_{e_{1}} and y_{0} ∈ X_{e_{n}} with d(x_{0}, \overline{xy}), d(y_{0}, \overline{xy}) ≤ C. By convexity of the distance function d_{X} it suffices to show that \overline{x_{0}y_{0}} ∩ B(p, N_{2}R) ≠ ∅.

Let K and β be as in Lemma 4.5 and (T , f, ϕ) be a K template for γ whose angles are bounded by β. Let α : \overline{x_{0}y_{0}} → T be the map guaranteed by Proposition 4.6. So Length(α) ≤ d(x_{0}, y_{0}) + nC_{5}. By part 3 of definition 4.4 there is a p′ ∈ T be such that d(p, ϕ(p′)) < R + K and hence (since X_{e_{i}} ⊂ N_{K}ϕ(W_{i})) d(ϕ(p′), ϕ(W_{i})) < 2R + 2K. Since, by Proposition 4.6, ϕ is an (L, A)-quasi isometric embedding we have B(p′, R_{2}) ∩ W_{i} ≠ ∅ for R_{2} = L(2R + 2K) + A. We will choose R_{0} and N_{2} large enough so that for n > N_{2} and R ≥ R_{0} we will have nC_{5} < C_{1}(n − C_{2})N_{1}R_{2} for the N_{1}(β), C_{1}(β) and C_{2}(β) of Proposition 5.2. Thus Proposition 5.2 forces α to intersect B(p′, N_{1}R_{2}). So we conclude (again using Proposition 4.6) that d(\overline{x_{0}y_{0}}, p) < C_{5} + d(ϕ(α), ϕ(p′)) + R + K ≤ C_{5} + LN_{1}R_{2} + A + R + K. The proposition now follows by taking N_{2} and R_{0} large enough.

5.2. Paths with small length distortion

Proposition 5.7. Pick M > 0 and α ∈ (\frac{1}{2}, 1]. Then there is a constant C = C(M, α) so that if 1 ≤ A ≤ B, η : [A, B] → X is a (not necessarily continuous) map to a Hadamard space X, and for all A ≤ t_{1} ≤ t_{2} ≤ B we have

|d_{X}(η(t_{1}), η(t_{2})) − (t_{2} − t_{1})| ≤ M(1 + log(\frac{t_{2}}{t_{1}} ))  (5.8)

then

d(η(t), z) ≤ C(1 + t^{α})  (5.9)

where z ∈ \overline{η(A)η(B)} is the point with d(z, η(A)) = \frac{t−A}{B−A}d(η(A), η(B)). Similarly, if A ≥ 1 and η : [A, ∞) → X satisfies (5.8) for all A ≤ t_{1} ≤ t_{2}, then there is a unique unit speed geodesic ray γ : [A, ∞) → X with γ(A) = η(A) such that

d_{X}(η(t), γ(t)) ≤ C(1 + t^{α})

for all t ∈ [A, ∞).

Proof. First note that we may assume that A = 1, since the map η_{1} : [1, B −A+1] → X given by η_{1}(t) := η(t + A − 1) will satisfy the hypotheses of the proposition, and the conclusion of the proposition applied to η_{1} will imply (5.9) for η.

Step 1: When 1 ≤ s_{1} ≤ s_{2} ≤ 2s_{1} ≤ B and s_{1} is sufficiently large then the comparison angle ˜∠_{η(1)}(η(s_{1}), η(s_{2})) ≤ Const s_{1}^{α−1}.

We will make use of the following lemma that follows from standard comparisons.

Lemma 5.10. Let X be a Hadamard space, x, y, z ∈ X. Set L := d(x, z), and the excess E := d(x, y) + d(y, z) − d(x, z). Then

d(y, \overline{xz}) ≤ \frac{\sqrt{2LE}}{2} \sqrt{1 + \frac{E}{2L}}  (5.11)

≤ \sqrt{LE} if E ≤ 2L.  (5.12)

Proof. Triangle comparison. Proof of Proposition 5.7 continued. Take 1 ≤ s_{1} ≤ s_{2} ≤ B, and consider the triple η(1), η(s_{1}), η(s_{2}). The excess for the triple is

≤ M(1 + log s_{2}) + M(1 + log s_{1}) + M(1 + log(\frac{s_{2}}{s_{1}} )) = 3M(1 + log s_{2}).

Since d(η(1), η(s_{2})) ≥ (s_{2} − 1) − M(1 + log s_{2}) when s_{2} > c_{1} = c_{1}(M), then the excess is ≤ 2d(η(1), η(s_{2})). Thus since d(η(1), η(s_{2})) ≤ (s_{2} − 1) + M(1 + log s_{2}) applying (5.12) we get

d(η(s_{1}), \overline{η(1)η(s_{2})}) ≤ \sqrt{[(s_{2} − 1 + M(1 + log s_{2}))][3M(1 + log s_{2})]}

≤ c_{2}(1 + s_{2}^{α})  (5.13)

where c_{2} = c_{2}(M, α). Therefore the comparison angle ˜∠_{η(1)}(η(s_{1}), η(s_{2})) satisfies

sin(˜∠_{η(1)}(η(s_{1}), η(s_{2}))) ≤ \frac{c_{2}(1 + s_{2}^{α})}{d(η(1), η(s_{1}))} ≤ \frac{c_{2}(1 + s_{2}^{α})}{[(s_{1} − 1) − M(1 + log s_{1})]}.

So there are constants c_{3} = c_{3}(M, α) and c_{4} = c_{4}(M, α) so that if c_{3} ≤ s_{1} ≤ s_{2} ≤ 2s_{1} ≤ B then

sin(˜∠_{η(1)}(η(s_{1}), η(s_{2}))) ≤ \frac{2c_{2}(1 + s_{2}^{α})}{s_{1}} ≤ \frac{c_{4}}{2} s_{2}^{α−1}  (5.14)

and

˜∠_{η(1)}(η(s_{1}), η(s_{2})) ≤ c_{4}s_{2}^{α−1}.  (5.15)

Step 2: Estimating d(η(t), \overline{η(1)η(B)}). Now pick t_{0} ∈ [1, B] with t_{0} ≥ c_{3} . Let t_{i} = 2^{i}t_{0} for i = 0, …, n where n is the integer part of \frac{log B}{log 2} , and t_{n+1} = B. Then \frac{B}{t_{n}} < 2. Applying the estimate (5.15) with u_{i} := η(t_{i}) we have for i = 1, …, n + 1:

˜∠_{η(1)}(u_{i−1}, u_{i}) ≤ c_{4}t_{i}^{α−1}

since \frac{t_{i}}{t_{i−1}} ≤ 2 and t_{i} ≥ c_{3}. Either n = 0, in which case we have

d(η(t_{0}), \overline{η(1)η(B)}) ≤ c_{2}(1 + t_{1}^{α}) by (5.13)
≤ c_{2}(1 + 2^{α}t_{0}^{α}) ≤ c_{5}(1 + t_{0}^{α})

where c_{5} = c_{5}(M, α). Otherwise, if t_{0} > c_{6} = c_{6}(M, α),

d(η(1), u_{0}) ≤ (t_{0} − 1) + M(1 + log t_{0})
≤ (t_{i} − 1) − M(1 + log t_{i}) ≤ d(η(1), u_{i}).

So for 0 ≤ i ≤ n + 1 we may pick v_{i} ∈ \overline{η(1)u_{i}} with d(η(1), v_{i}) = d(η(1), u_{0}) =: R_{0}. By triangle comparison we have

d(v_{i−1}, v_{i}) ≤ R_{0}˜∠_{η(1)}(v_{i−1}, v_{i}) ≤ R_{0}˜∠_{η(1)}(u_{i−1}, u_{i})
≤ R_{0}c_{4}t_{i}^{α−1}

so

d(u_{0}, \overline{η(1)η(B)}) = d(u_{0}, \overline{η(1)u_{n+1}}) ≤ d(u_{0}, v_{n+1})

≤ ∑_{i=1}^{n+1} d(v_{i−1}, v_{i}) ≤ R_{0}c_{4} ∑_{i=1}^{n+1} t_{i}^{α−1}

≤ R_{0}c_{7}t_{0}^{α−1}

for c_{7} = c_{7}(M, α) (independent of n). We have R_{0} ≤ t_{0} − 1 + M(1 + log t_{0}) ≤ 2t_{0} when t_{0} is sufficiently large, so when t_{0} ≥ c_{8} = c_{8}(M, α)

d(u_{0}, \overline{η(1)η(B)}) ≤ c_{9}t_{0}^{α}

where c_{9} = c_{9}(M, α).

Step 3: Estimating d(η(t), z) where z ∈ \overline{η(1)η(B)} satisfies d(z, η(1)) = \frac{t−1}{B−1}d(η(1), η(B)). Let x ∈ \overline{η(1)η(B)} be the point nearest η(t). Then for t ≥ c_{10} we have

d(η(t), z) ≤ d(η(t), x) + d(x, z)

= d(η(t), x) + |d(x, η(1)) − (t − 1)\frac{d(η(1), η(B))}{B − 1}|

≤ d(η(t), x) + [d(x, η(t)) + \frac{(t − 1)}{B − 1}M(1 + log B) + M(1 + log t)] ≤ 3c_{9}t^{α}.

Thus the result holds for large t, but by choosing C large enough we get the result for all t.

For the ray case we redo step 3 above when d(η(1), η(B)) > t for z_{1} ∈ \overline{η(1)η(B)} the point which satisfies d(z_{1}, η(1)) = t. Let x ∈ \overline{η(1)η(B)} be the point nearest η(t). Then for t ≥ c_{11} we have

d(η(t), z_{1}) ≤ d(η(t), x) + d(x, z_{1})
= d(η(t), x) + |d(x, η(1)) − (t − 1)|
≤ d(η(t), x) + [d(x, η(t)) + M(1 + log t)] ≤ 3c_{9}t^{α}.

Now choose γ as a limit of a subsequence of \overline{η(1), η(B)} as B goes to ∞ (the result will in fact imply that the sequence itself converges). Since none of the constants depended on B the above estimate for d(η(t), z_{1}) gives the result for rays.

5.3. The proof of Theorem 5.1

Let β = β(X) be the minimum angle between singular geodesics of a template for X defined in Lemma 4.5, and let C_{2}(β), N_{1}(β) be the constants from Corollary 5.5. Let N_{2}, R_{0} be the constants from Proposition 5.6; we will assume that N_{2} ≥ max(C_{2}(β), N_{1}(β)).

Definition 5.16. For every ψ > 0 , we will say that z ∈ \overline{xy} is a ψ-cluster point if the segment \overline{B(z, ψd(z, x))} ∩ \overline{xy} intersects at least N_{2} walls of T , or if z ∈ \{x, y\}. We will let P_{ψ} ⊂ \overline{xy} represent the set of ψ-cluster points.

Proof of 1. Let (T , f, ϕ), x, y, be as in the statement of part 1 of the theorem. We begin with two lemmas.

Lemma 5.17. There is a constant c_{1} depending on K such that if \overline{xy} intersects a wall W ∈ W all_{T}, then \overline{ϕ(x)ϕ(y)} ∩ N_{c_{1}}(X_{f(W)}) ≠ ∅.

Proof. If {x, y}∩W ≠ ∅ this is immediate since ϕ(W) ⊂ N_{K}(X_{f(W)}) by definition 4.4. Otherwise x and y must lie in distinct components of T −W because each component is convex. Say x ∈ W_{1} ∪ S_{1} ∪ W′_{1} and y ∈ W_{2} ∪ S_{2} ∪ W′_{2} where W_{i} is adjacent to W′_{i}. Set v := f(W_{1}) ∩ f(W′_{1}) and v′ := f(W_{2}) ∩ f(W′_{2}). Applying Lemma 3.17 the lemma follows.

Lemma 5.18. Suppose p_{1}, p_{2} ∈ \overline{xy}, d(p_{1}, x) ≤ d(p_{2}, x), and \overline{p_{1}p_{2}} ∩ P_{ψ} ⊂ \{p_{1}, p_{2}\}. Set A := 1 + d(p_{1}, x), B := 1 + d(p_{2}, x). Let η_{0} : [A, B] → T be the unit speed parameterization of \overline{p_{1}p_{2}}. Then the composition η := ϕ ∘ η_{0} : [A, B] → X satisfies the hypotheses of Proposition 5.7 with M = M(K, ψ).

Proof. Pick A ≤ t_{1} < t_{2} ≤ B. By Corollary 4.10 we have

|d_{X}(η(t_{1}), η(t_{2})) − (t_{2} − t_{1})| ≤ kC_{6}  (5.19)

where \overline{η_{0}(t_{1})η_{0}(t_{2})} intersects at most (k − 1) walls and strips of T . For each z ∈ \overline{η_{0}(t_{1})η_{0}(t_{2})}, the segment \overline{B(z, ψd(z, x))} ∩ \overline{xy} intersects at most N_{2} − 1 walls. We can cover \overline{η_{0}(t_{1})η_{0}(t_{2})} − B(x, 1) with at most Const log(\frac{t_{2}}{t_{1}} ) such segments, and \overline{B(x, 1)} ∩ \overline{η_{0}(t_{1})η_{0}(t_{2})} intersects at most 2 walls (since strips have width at least 1), so the lemma follows from (5.19).

Step 1: Estimating d(ϕ(z), \overline{ϕ(x)ϕ(y)}) when z ∈ P_{ψ}. Suppose z ∈ P_{ψ}. Then \overline{xy} ∩B(z, ψR) intersects at least N_{2} consecutive walls W_{1}, …, W_{n} of T . By definition 4.4 and the fact that ϕ : T → X is a quasi-isometric embedding (Proposition 4.6) there is a constant c_{2} = c_{2}(K) so that d_{X}(ϕ(z), X_{f(W_{i})}) ≤ c_{2}(1 + ψR) for i = 1, … n. By Lemma 5.17 and Proposition 5.6, we conclude that \overline{ϕ(x)ϕ(y)} ∩ B(ϕ(z), N_{2}c_{2}(1 + ψR) + 2c_{1}) ≠ ∅ provided c_{2}(1 + ψR) ≥ R_{0}. So there are positive constants r_{0} = r_{0}(K, ψ), c_{3} = c_{3}(K, β) so that if z ∈ P_{ψ} and R := d(z, x) ≥ r_{0}, then

d(ϕ(z), \overline{ϕ(x)ϕ(y)}) ≤ c_{3}ψR.  (5.20)

Hence for any z ∈ P_{ψ}

d(ϕ(z), \overline{ϕ(x)ϕ(y)}) ≤ c_{3}ψR + Lr_{0} + A = c_{3}ψR + c_{4}  (5.21)

where (L, A) are the quasi-isometric embedding constants of ϕ, and c_{4} := Lr_{0} + A = c_{4}(K, ψ).

Step 2: Estimating d(ϕ(z), \overline{ϕ(x)ϕ(y)}) when z ∉ P_{ψ}. Pick z ∉ P_{ψ}. There are points p_{1}, p_{2} ∈ \overline{xy} so that z ∈ \overline{p_{1}p_{2}}, d(p_{1}, x) ≤ d(p_{2}, x), and either \overline{p_{1}p_{2}} ∩ P_{ψ} = \{p_{1}, p_{2}\} or one or both of p_{1} = x or p_{2} = y hold. Each step of the argument below holds in the special cases p_{1} = x or p_{2} = y (often for easier reasons) so we will ignore them. By (5.21) we have

d(ϕ(p_{i}), \overline{ϕ(x)ϕ(y)}) ≤ c_{3}ψd(p_{i}, x) + c_{4}  (5.22)

for i = 1, 2. Since p_{1}, p_{2} satisfy the conditions of Lemma 5.18, we may apply Proposition 5.7 with α = \frac{3}{4} to get that for the w ∈ \overline{ϕ(p_{1})ϕ(p_{2})} with d(w, ϕ(p_{1})) = \frac{d(z,p_{1})}{d(p_{1},p_{2})}d(ϕ(p_{1}), ϕ(p_{2})) we have

d(ϕ(z), w) ≤ C(1 + [d(z, x)]^{\frac{3}{4}} ) ≤ ψd(z, x) + c_{6}.  (5.23)

Where c_{6} = c_{6}(K, ψ).

On the other hand for ζ = \frac{d(w,ϕ(p_{1}))}{d(ϕ(p_{1}),ϕ(p_{2}))} = \frac{d(z,p_{1})}{d(p_{1},p_{2})} the convexity of the distance function d(∗, \overline{ϕ(x)ϕ(y)}) and 5.22 says that

d(w, \overline{ϕ(x)ϕ(y)}) ≤ c_{3}ψ((1 − ζ)d(p_{1}, x) + ζd(p_{2}, x)) + c_{4} = c_{3}ψd(z, x) + c_{4} (5.24)

Combining (5.24) and (5.23) we get

d(ϕ(z), \overline{ϕ(x)ϕ(y)}) ≤ c_{7}ψd(z, x) + c_{8}  (5.25)

for c_{7} = c_{7}(K) and c_{8} = c_{8}(K, ψ). Thus (5.21) and (5.25) together imply that for every choice of ψ > 0 there are constants c_{9}(K, β) and c_{10}(K, ψ) such that for all z

d(ϕ(z), \overline{ϕ(x)ϕ(y)}) ≤ c_{9}ψR + c_{10}.

The fact that c_{9} does not depend on ψ allows us to pick θ(R) which decays to 0 as R → ∞ such that for each R there is a ψ = ψ(R) > 0 (for example choose ψ(R) decaying to 0 such that c_{10}(ψ) < R^{− \frac{1}{2}} ) so that

c_{9}ψR + c_{10}(ψ) ≤ (1 + R)θ(R)

This implies part 1 of Theorem 5.1.

Proof of 2. We omit this, as it is similar to the proof of 1.

Proof of 3. Part 3 follows directly from part 1 and Lemma 2.5.

Proof of 4. Part 2 along with reasoning similar to the proof of Lemma 2.5 shows that if x, y_{k} ∈ T , ξ ∈ ∂_{∞}X, and \overline{ϕ(x)ϕ(y_{k})} → \overline{xξ}, then \overline{xy_{k}} converges to some ray \overline{xξ′} ⊂ T . Hence ∂_{∞}ϕ is a homeomorphism from ∂_{∞}T onto the limit set of ϕ(T ) ⊂ X, which we denote by ∂_{∞}ϕ(T ). It follows immediately from this that ∂_{∞}ϕ maps ∂_{∞}T −∂_{∞}^{∞}T = ∪_{W ∈W all_{T}} ∂_{∞}W homeomorphically onto ∪_{W ∈W all_{T}} ∂_{∞}X_{f(W)}. This implies 4 when γ ⊂ T is a geodesic segment, so we now assume that γ is a geodesic ray, and we need only show that ∂_{∞}ϕ(∂_{∞}^{∞}T ) = ∂_{∞}^{η}X where η = ∂_{∞}γ ∈ ∂_{∞}T. We will let W_{k} ∈ W all_{T} be the k^{th} wall of T .

We first show ∂_{∞}ϕ(∂_{∞}^{∞}T ) ⊂ ∂_{∞}^{η}X. Suppose ξ ∈ ∂_{∞}^{∞}T , z_{k} ∈ \overline{xξ} ∩ W_{k}, and d(z_{k}, x) → ∞. Thus \overline{ϕ(x)ϕ(z_{k})} → \overline{ϕ(x)∂_{∞}ϕ(ξ)}. By lemma 3.17 part 1, for any l we have \overline{ϕ(x)ϕ(z_{k})} ∩ N_{c}(X_{f(W_{l})}) ≠ ∅ for sufficiently large k. So by Lemma 2.4 either ∂_{∞}ϕ(ξ) ∈ ∂_{∞}N_{c}(X_{f(W_{l})}) = ∂_{∞}X_{f(W_{l})} or \overline{ϕ(x)∂_{∞}ϕ(ξ)} ∩ N_{c}(X_{f(W_{l})}) ≠ ∅. The former case is impossible since we already know that (∂_{∞}ϕ)^{−1}(∂_{∞}X_{f(W_{l})}) = ∂_{∞}W_{l}. So \overline{ϕ(x)∂_{∞}ϕ(ξ)} ∩ N_{c}(X_{f(W_{l})}) ≠ ∅ for every W ∈ W all_{T}, forcing I(\overline{ϕ(x)∂_{∞}ϕ(ξ)}) = γ. So ∂_{∞}ϕ(∂_{∞}^{∞}T ) ⊂ ∂_{∞}^{η}X.

We now show ∂_{∞}^{η}X ⊂ ∂_{∞}ϕ(∂_{∞}^{∞}T ). Suppose x ∈ W_{1} ∈ W all_{T} and ξ ∈ ∂_{∞}^{η}X. Then from the definition of itineraries 3.20 and Lemma 3.19 there are z_{k} ∈ \overline{ϕ(x)ξ} ∩N_{C}(X_{e_{k}}) for e_{k} ∈ T so that d_{T} (e_{k}, f(W_{1})) → ∞ and d_{T} (e_{k}, γ) is uniformly bounded. So for all but finitely many W, f(W) separates e_{k} from f(W_{1}) for sufficiently large k, so by Lemma 3.17

\overline{ϕ(x)z_{k}} ∩ N_{c}(X_{f(W)}) ≠ ∅

for sufficiently large k. Hence ξ belongs to the limit set of ∪_{W ∈W all_{T}} X_{f(W)}, which is the same as ∂_{∞}(ϕ(T )) = ∂_{∞}ϕ(∂_{∞}T ). Therefore ξ ∈ (∂_{∞}ϕ)(∂_{∞}T − ∪_{W ∈W all_{T}} ∂_{∞}W) = (∂_{∞}ϕ)(∂_{∞}^{∞}T ).

Proof of 5. Pick x ∈ T and ξ ∈ ∂_{∞}^{∞}T . Suppose there is a sequence z_{k} ∈ \overline{xξ} with lim_{k→∞}(z_{k}, x) = ∞ so that z_{k} is a ψ_{k}-cluster point \overline{xξ} where ψ_{k} → 0. Set R_{k} := d(z_{k}, x). Applying Corollary 5.5 to each z_{k}, we see that if ξ′ ∈ ∂_{∞}^{∞}T , then for sufficiently large k the intersection \overline{xξ′} ∩ B(z_{k}, N_{1}ψ_{k}R_{k}) is nonempty, and this clearly forces \overline{xξ′} = \overline{xξ}. In this case we have ∂_{∞}^{∞}T = \{ξ\}, and 5 is immediate. So we may assume that for every ξ_{1}, ξ_{2} ∈ ∂_{∞}^{∞}T there is a ψ > 0 such that \overline{xξ_{1}} and \overline{xξ_{2}} contain no ψ-cluster points. Let η̄_{i} : [1, ∞) → T be the unit speed parameterization of \overline{xξ_{i}}, and let η_{i} : [1, ∞) → X be the composition ϕ ∘ η̄_{i} : [1, ∞) → X. Then by Lemma 5.18 the η_{i} satisfy the hypotheses of Proposition 5.7 for a suitable M; so we have unit speed geodesic rays γ_{i} : [1, ∞) → X with γ_{i}(1) = η_{i}(1) and d_{X}(η_{i}(t), γ_{i}(t)) ≤ C(1 + t^{\frac{3}{4}} ). Now, for sufficiently large k, choose t_{k}^{i} so that η̄_{i}(t_{k}^{i}) lies in the k^{th} wall of T . By definition 4.4

|d_{X}(η_{1}(t_{k}^{1}), η_{2}(t_{k}^{2})) − d_{T} (η̄_{1}(t_{k}^{1}), η̄_{2}(t_{k}^{2}))| < K.

Thus

∠_{T} (γ_{1}, γ_{2}) = lim_{k→∞} ˜∠_{η_{1}(1)}(η_{1}(t_{k}^{1}), η_{2}(t_{k}^{2})) = lim_{k→∞} ˜∠_{x}(η̄_{1}(t_{k}^{1}), η̄_{2}(t_{k}^{2})) = ∂_{T} (ξ_{1}, ξ_{2}).

This proves 5.

5.4. Applications of Theorem 5.1

Theorem 5.1 has a number of corollaries:

Corollary 5.26. Let G be the fundamental group of an admissible graph of groups G, and let G ↷ T be the Bass-Serre tree of G. Let γ ⊂ T be a geodesic ray with i^{th} edge e_{i} ⊂ T, and set η := ∂_{∞}γ ∈ ∂_{∞}T. Then for any admissible action G ↷ X, the subset ∂_{∞}^{η}X ⊂ ∂_{∞}X defined in section 3.3 is precisely the set of limit points of the sequence of subsets ∂_{∞}X_{e_{k}} ⊂ ∂_{∞}X.

Proof. Let (T , f, ϕ) be a template for γ ⊂ T. By parts 3 and 4 of Theorem 5.1, ∂_{∞}ϕ : ∂_{∞}T → ∂_{∞}X is a homeomorphism onto (∪_{k} ∂_{∞}X_{e_{k}}) ∪ (∂_{∞}^{η}X). Therefore it suffices to show that ∂_{∞}^{∞}T ⊂ ∂_{∞}T is the set of limit points of the sequence ∂_{∞}f^{−1}(e_{k}) ⊂ ∂_{∞}T . To see this, observe that any geodesic segment \overline{px} ⊂ T which arrives at a wall W ∈ W all_{T} via a strip S ∈ Strip_{T} may be prolonged by a geodesic ray contained in W; this implies that every ξ ∈ ∂_{∞}^{∞}T is a limit of a sequence ξ_{k} ∈ ∂_{∞}f^{−1}(e_{k}). On the other hand, if ξ_{k} ∈ ∂_{∞}f^{−1}(e_{k}) converges to ξ ∈ ∂_{∞}T , then Lemma 2.4 implies that either ξ ∈ ∂_{∞}^{∞}T , or ξ belongs to ∂_{∞}f^{−1}(e_{k}) for all sufficiently large k, which is absurd.

Remark 5.27. Let G ↷ X, T, γ, and η be as in Corollary 5.26. When X is a 3-dimensional Hadamard manifold, then ∂_{∞}X ≃ S^{2} and there is an alternate characterization of ∂_{∞}^{η}X which uses little more than the definition of itineraries. For each i, ∂_{∞}X_{e_{i}} ⊂ ∂_{∞}X ≃ S^{2} determines two closed disks by the Jordan separation theorem; let D_{i} be the one which contains ∂_{∞}X_{e_{j}} for all j ≥ i. Then ∂_{∞}^{η}X = ∩_{i}D_{i}. To see this note that if F ⊂ X is a flat totally geodesic plane and p ∈ X − F, then the two components of ∂_{∞}X −∂_{∞}F are \{ξ ∈ ∂_{∞}X | \overline{pξ} ∩F ≠ ∅\} and \{ξ ∈ ∂_{∞}X | \overline{pξ} ∩F = ∅\}.

Corollary 5.28. Let G ↷ X and G ↷ X′ be admissible actions, let G ↷ T be the Bass-Serre tree of G, and let V, E be the sets of vertices and edges of T, respectively. If Φ : ∂_{∞}X → ∂_{∞}X′ is any G-equivariant homeomorphism, then

1. Φ maps ∂_{∞}X_{σ} homeomorphically to ∂_{∞}X′_{σ} for all σ ∈ V ∪ E.

2. Φ maps ∂_{∞}^{η}X homeomorphically to ∂_{∞}^{η}X′ for all η ∈ ∂_{∞}T.

Proof. Part 1 follows from the characterization of ∂_{∞}X_{σ} as a fixed point set which is stated in Lemma 3.23. Part 2 follows from part 1 and Corollary 5.26.

Corollary 5.29. Let G ↷ X be an admissible action, and G ↷ T be the Bass-Serre action for G. Then

1. The union ∪_{v∈V} ∂_{T} X_{v} ⊂ ∂_{T} X is a CAT (1) space with respect to the induced metric, and may be described metrically as follows. First, ∂_{T} X_{v} is a metric suspension of an uncountable discrete CAT (1) space for each v ∈ V, and ∂_{T} X_{e} is isometric to the standard circle for every e ∈ E. Take the disjoint union ⨿_{v∈V} ∂_{T} X_{v}, and for each edge e = \overline{v_{1}v_{2}} ∈ E, glue ∂_{T} X_{v_{1}} to ∂_{T} X_{v_{2}} isometrically by identifying the copies of ∂_{T} X_{e} ⊂ ∂_{T} X_{v_{i}}; the result is isometric to ∪_{v∈V} ∂_{T} X_{v} ⊂ ∂_{T} X.

2. The union ∪_{v∈V} ∂_{T} X_{v} forms a connected component of ∂_{T} X. The remaining components are contained in the subsets ∂_{T}^{η} X for η ∈ ∂_{∞}T. We will show in Lemma 7.3 that each ∂_{T}^{η} X is either a point or isometric to an interval of length < π.

Proof. To prove 1, we first observe that if e_{1}, …, e_{n} is an edge path in T with initial vertex v_{1} and terminal vertex v_{n}, ξ_{1} ∈ ∂_{T} X_{v_{1}}, ξ_{n} ∈ ∂_{T} X_{v_{n}}, and ∠_{T} (ξ_{1}, ξ_{n}) < π, then the Tits segment \overline{ξ_{1}ξ_{n}} ⊂ ∂_{T} X is contained in ∪_{i=1}^{n} ∂_{T} X_{v_{i}}. To see this, pick ξ ∈ \overline{ξ_{1}ξ_{n}}. Recall that for a given base point x, \overline{xξ} may be obtained as the limit of a sequence \overline{xy_{k}} where y_{k} lies on a segment \overline{x_{1}^{k}x_{2}^{k}} and x_{i}^{k} ∈ \overline{xξ_{i}} is a sequence tending to infinity. The quasi-convexity property of Lemma 4.3 (applied in sucession to \overline{xξ_{i}}, \overline{x_{1}^{k}x_{2}^{k}}, and then \overline{xy_{k}}) implies that \overline{xy_{k}} ⊂ N_{C}(∪_{i=1}^{n}X_{v_{i}}) for some C, and the convexity of the N_{C}(X_{v_{i}})’s implies that ξ ∈ ∂_{T} X_{v_{i}} for some i ∈ {1, … n}. Part 1 now follows from Corollary 3.16.

Before proving 2, we recall that open balls of radius \frac{π}{2} in CAT (1) spaces are geodesically convex, so two points in a CAT (1) space belong to the same connected component iff they can be joined by a unit speed path.

Suppose η ∈ ∂_{∞}T and ξ ∈ ∂_{T}^{η} X. Fix v ∈ V, p ∈ X_{v}, and let e_{k} ∈ E denote the k^{th} edge of the ray \overline{vη} ⊂ T. Part 1 of Lemma 3.19 implies that \overline{pξ} ∩ X_{e_{k}} ≠ ∅ for all but finitely many k. If c : [0, L] → ∂_{T} X is a unit speed path starting at ξ, then by Lemma 2.4, we find that either c([0, L]) ⊂ ∂_{T}^{η} X or for all sufficiently large k there is a t_{k} ∈ [0, L] so that \overline{pc(t_{k})} ∈ ∂_{T} X_{e_{k}}. But part 1 shows that when e, e′ ∈ E and d(e, e′) ≥ 2, then d(∂_{T} X_{e}, ∂_{T} X_{e′}) is bounded away from 0, which gives a contradiction. Therefore c([0, L]) ⊂ ∂_{T}^{η} X and we have shown that the connected component of ξ is contained in ∂_{T}^{η} X.

Finally, we note that ∪_{v∈V} ∂_{T} X_{v} is connected: if v_{1}, …, v_{k} are the consecutive vertices of a geodesic segment in T, then ∂_{T} X_{v_{i}} ∩ ∂_{T} X_{v_{i+1}} = ∂_{T} X\overline{v_{i}v_{i+1}} ≠ ∅.

6. Geometric data and equivariant quasi-isometries

Throughout this section, G ↷^{ρ} X and G ↷^{ρ′} X′ will denote admissible actions of an admissible group G on Hadamard spaces X and X′, and we let MLS_{v}, τ_{v} and MLS′_{v}, τ′_{v} denote their respective geometric data (see Definition 3.9). The main result in this section is Theorem 6.5, which shows that a G-equivariant quasi-isometry Φ : X → X′ induces an equivariant homeomorphism ∂_{∞}Φ : ∂_{∞}X → ∂_{∞}X′ provided ρ and ρ′ have equivalent geometric data.

Definition 6.1. We say that ρ and ρ′ have equivalent geometric data if there are functions λ : V → ℝ_{+} and μ : V → ℝ_{+} so that for every v ∈ V

MLS′_{v} = λ(v)MLS_{v} and τ′_{v} = μ(v)τ_{v}.

It follows from the G-invariance of the geometric data that λ and μ will be G-invariant.

The structure of G strongly restricts the possibilities for the functions λ and μ:

Lemma 6.2. Suppose the geometric data for the actions G ↷ X and G ↷ X′ are equivalent, and let λ : V → ℝ_{+} and μ : V → ℝ_{+} be as in Definition 6.1. Then either

1. λ ≡ μ ≡ a, for some a > 0.

or

2. There are constants a and b, a ≠ b, so that λ(V) = {a, b} = μ(V). Moreover, for any pair v_{1}, v_{2} of adjacent vertices in T, λ(v_{1}) = μ(v_{2}), λ(v_{2}) = μ(v_{1}), and the ℝ directions of Y_{v_{i}} ≃ Ȳ_{v_{i}} × ℝ determine orthogonal directions in Y_{e}, where e := \overline{v_{1}v_{2}}. In particular, there is a G-equivariant 2-coloring of V (i.e. a two coloring of the finite graph G = T/G) such that λ and μ are functions of the vertex color.

Proof. Pick e = \overline{v_{1}v_{2}} ∈ E, and consider G_{e} ⊗ ℝ ≃ ℤ^{2} ⊗ ℝ ≃ ℝ^{2}. For i = 1, 2 we have subspaces Z_{i} := (Z(G_{v_{i}}) ∩ G_{e}) ⊗ ℝ ≃ ℤ ⊗ ℝ ≃ ℝ determined by the centers of the G_{v_{i}}’s. The action G_{e} ↷ Y_{e} induces an inner product ⟨·, ·⟩ on G_{e} ⊗ ℝ by letting, for g ∈ G_{e}, ⟨g, g⟩ = δ_{g}^{2}. (For p ∈ Y_{e} there is an embedded Euclidean plane ℝ^{2} ⊂ Y_{e} invariant under the action of G_{e} ↷ Y_{e} on which G_{e} acts by translations of δ_{g}. So our metric is naturally related to the metric on this ℝ^{2}). We can extend the maps

[figure: Z_{1}; Z_{2}; Ker(τ̂_{1}); Ker(τ̂_{2})]

Figure 1: On G_{e} ⊗ ℝ ≃ ℝ^{2} we have two metrics < ·, · > and < ·, · >′. Ker(τ̂_{i}) (which is the same as Ker(τ̂′_{i})) for i ∈ {1, 2} is perpendicular to Z_{i} with respect to both metrics.

τ_{i} : G_{e} → R to linear maps τ̂_{i} : G_{e} ⊗ ℝ → R. For g ∈ G_{e} we see that τ_{i}(g) is just the change in the i-vertical component from y to g(y). So for x ∈ G_{e} ⊗ ℝ, τ̂_{i}(x) is just the length of the orthogonal projection of x onto Z_{i}. In particular, for x ∈ Z_{i} we have ⟨x, x⟩ = τ̂_{i}(x)^{2}, and Ker(τ̂_{i}) is ⟨·, ·⟩ perpendicular to Z_{i}. Similarly M̂LS_{i}(x) corresponds to the length of the projection of x perpendicular to Z_{i} (i.e. to Ker(τ_{i})). In general Z_{1} and Z_{2} are not perpendicular but they are always linearly independent (see figure 1.

Similarly, using the action G_{e} ↷ Y′_{e}, we get an induced inner product ⟨·, ·⟩′ and linear maps τ̂′_{1} and τ̂′_{2}. By assumption τ̂′_{i} = μ(v_{i})τ̂_{i}, hence Ker(τ̂_{i}) = Ker(τ̂′_{i}). So the space perpendicular to Z_{1} (resp. Z_{2}) with respect to ⟨·, ·⟩ is the same as that with respect to ⟨·, ·⟩′. Hence, by the independence of Z_{1} and Z_{2}, either ⟨·, ·⟩′ = a⟨·, ·⟩ for some a > 0 or Z_{1} and Z_{2} are perpendicular in both metrics so Ker(τ̂_{1}) = Ker(τ̂′_{1}) = Z_{2} and Ker(τ̂_{2}) = Ker(τ̂′_{2}) = Z_{1}. In the latter case, choosing x ∈ Z_{2} we have ⟨x, x⟩′ = (M̂LS_{1}(x))^{2} = μ(v_{2})^{2}(M̂LS_{1}(x))^{2} = μ(v_{2})^{2}⟨x, x⟩ and ⟨x, x⟩′ = (τ′_{2}(x))^{2} = λ(v_{1})^{2}(τ_{2}(x))^{2} = λ(v_{1})^{2}⟨x, x⟩ so λ(v_{1}) = μ(v_{2}); similarly λ(v_{2}) = μ(v_{1}).

We now fix a G-equivariant (L, A)-quasi-isometry Φ : X → X′ for the rest of this

section. The constants defined will depend on the geometry of G ↷^{ρ} X and G ↷^{ρ′} X′ as well as any other explicitly stated quantities.

Lemma 6.3. There is a constant D_{1} = D_{1}(L, A) so that for every σ ∈ V ∪ E, the Hausdorff distance d_{H}(Φ(X_{σ}), X′_{σ}) is at most D_{1}.

Proof. For g ∈ G let d_{g} and d′_{g} denote the displacement functions for g in X and X′ respectively. In particular d′_{g}(x) ≤ Ld_{g}(x) + A.

For any v ∈ V, if g ∈ Z(G_{v}) is a generator of the center Z(G_{v}) then (see section 2.3) d_{g} is proper on X/Z(Z(G_{v}), G) = X/G_{v} (by Lemma 3.6) and hence d_{g} (resp d′_{g}) grows with the distance from X_{v} (resp X′_{v}). This (along with the fact that there are only finitely many σ modulo G) implies the lemma when σ ∈ V. If e ∈ E, and g_{1}, g_{2} ∈ G_{e} are a basis for G_{e} then max{d_{g_{1}}, d_{g_{2}}} (resp max{d′_{g_{1}}, d′_{g_{2}}}) grows with the distance from X_{e} (resp X′_{e}). This implies the lemma when σ ∈ E.

We now assume for the remainder of this section that ρ and ρ′ have equivalent geometric data, and we let λ : V → ℝ_{+} and μ : V → ℝ_{+} be as in Definition 6.1. For each v ∈ V, we have nearest point projections p_{v} : X → Y_{v} and p′_{v} : X′ → Y′_{v}. Modulo renormalization of the metrics on the spaces Y_{v}, the equivariant quasi-isometry Φ restricts to a Hausdorff approximation:

Lemma 6.4. For every v ∈ V, the G_{v}-equivariant quasi-isometry Φ_{v} := p′_{v} ∘ Φ|_{Y_{v}} : Y_{v} → Y′_{v} has the following properties:

1. It is at distance < D_{2} = D_{2}(L, A) from a G_{v}-equivariant map Ψ_{v} : Y_{v} → Y′_{v} which respects the product structures Y_{v} ≃ Ȳ_{v} × ℝ and Y′_{v} ≃ Ȳ′_{v} × ℝ.

2. If we stretch the metric on the Ȳ_{v} factor of Y_{v} by λ(v), and the metric on the ℝ factor by μ(v), then Φ_{v} becomes a D_{3} = D_{3}(L, A)-Hausdorff approximation, and maps unit speed geodesic segments to within D_{3} of unit speed geodesic segments.

Proof. We first prove part 1. The ℝ fibers of Y_{v} and Y′_{v} are within uniform Hausdorff distance of Z(G_{v})-orbits, so G_{v}-equivariance implies that Φ_{v} takes ℝ fibers of Y_{v} to within uniform Hausdorff distance (say C_{1}) of ℝ fibers of Y′_{v}. The Ȳ_{v} fibers of Y_{v} are within uniform Hausdorff distance of sets of the form {g(p) | p ∈ Y_{v},g ∈ G_{v}, |τ_{v}(g)| < C} for sufficiently large C, and a similar characterization of the Ȳ′_{v} fibers of Y′_{v} holds. We now define the product map Ψ_{v} = Ψ̄_{v} ×Ψ_{v}^{ℝ} : Ȳ_{v} ×ℝ → Ȳ′_{v}×ℝ. Fix a basepoint p ∈ Y_{v}. We may assume that the ℝ factor of p and Φ_{v}(p) are 0 and take Ψ_{v}^{ℝ}(t) = μ(v)t. We let S ⊂ Ȳ_{v} be a (set theoretical) cross section for the H_{v} action. For s ∈ S choose a g ∈ G_{v} such that |τ_{v}(g)| < C and d(g(p), (s, 0)) ≤ C_{1}. Now, since |τ′_{v}(g)| < μ(v)C, we can choose a point Ψ̄_{v}(s) such that d(g(Φ(p)), (Ψ̄_{v}(s), 0)) ≤ μ(v)C. We note that

d(Ψ((s, 0)), Φ((s, 0))) ≤ d((Ψ̄_{v}(s), 0), Φ(g(p))) + LC_{1} + A ≤ μ(v)C + LC_{1} + A

Extend this to an H_{v} equivariant map Ψ̄_{v} : Ȳ_{v} → Ȳ′_{v}. Thus, along with the fact that τ′(g) = μ(v)τ(g), we see that Ψ_{v} is a G_{v}-equivariant map. Now for every q ∈ Y_{v} there is a g ∈ G_{v} such that g(q) = (s, t) for some s ∈ S and some −C < t < C. Now d(Ψ(q), Φ(q)) =

d(Ψ((s, t)), Φ((s, t))) ≤ 2Cμ(v) + d(Ψ((s, 0)), Φ((s, 0))) ≤ 3μ(v)C + LC_{1} + A

This proves 1.

Part 2 follows from part 1 if we can show that Ψ̄_{v} and Ψ_{v}^{ℝ} carry unit speed geodesics to within uniform distance of unit speed geodesics. The map Ψ_{v}^{ℝ} clearly does, because the translation distance in the ℝ-direction is measured by τ_{v} (resp. τ′_{v}) and these have ratio μ(v). Recall that H_{v} := G_{v}/Z(G_{v}) acts discretely and cocompactly on the hyperbolic metric spaces Ȳ_{v} and Ȳ′_{v}. Since MLS′_{v} = λ(v)MLS_{v}, if we renormalize the metric on Ȳ_{v} by λ(v) we can apply Lemma 2.20 to see that Ψ̄_{v} : Ȳ_{v} → Ȳ′_{v} preserves unit speed geodesics up to uniform error. This proves 2.

We may now use our quasi-isometry Φ : X → X′ to transport standard Ktemplates for X to templates for X′ (see section 4.2). Start with a standard Ktemplate (T , f, ϕ) for some segment or ray γ ⊂ T. Recall that the walls of T come from flats F_{e} ⊂ Y_{e} ⊂ X_{e}. To produce the new template T′ distort the metric on T by an affine change as follows. For each v ∈ V, we think of scaling the metric on Ȳ_{v} by λ(v) and the ℝ-factor of Y_{v} by μ(v); and then we distort the flats F_{e} = F\overline{vv′} ⊂ F_{v} ∩F_{v′} and strips S_{ee′} ⊂ Y_{e∩e′} used to build T accordingly. Lemmas 6.3 and 6.4 imply that (T′, f, Φ ∘ ϕ) is a K′-template for γ where K′ = K′(L, A). Notice (using Lemma 6.2) that the identity map T → T′ is an affine map (i.e. maps constant speed geodesics to constant speed geodesics), and is a homothety when λ = μ = a ∈ ℝ.

Theorem 6.5. Let G ↷^{ρ} X and G ↷^{ρ′} X′ be admissible actions of an admissible group G on Hadamard spaces X and X′ such that ρ and ρ′ have equivalent geometric data, and let Φ : X → X′ be a G-equivariant (L, A)-quasi-isometry. Then there is a function θ : ℝ_{+} → ℝ_{+} (depending on K,L, A and the geometry of X and X’) with lim_{r→∞} θ(r) = 0 so that for every x, y ∈ X, z ∈ \overline{xy}, we have

d_{X′}(Φ(z), \overline{Φ(x)Φ(y)}) ≤ (1 + d_{X}(z, x))θ(d_{X}(z, x)).  (6.6)

Consequently, by Lemma 2.5, Φ extends to a unique map Φ̄ : X̄ → X̄′ which is continuous at ∂_{∞}X ⊂ X̄. Setting ∂_{∞}Φ := Φ̄|_{∂_{∞}X} : ∂_{∞}X → ∂_{∞}X′, we obtain a Gequivariant homeomorphism. If λ = μ = a ∈ ℝ, then ∂_{∞}Φ is an isometry with respect to Tits metrics.

Proof. Pick x, y ∈ X, and then find a segment γ ⊂ T so that x ∈ X_{e_{1}}, y ∈ X_{e_{n}}, and the i^{th} edge of γ is e_{i}. Now let (T , f, ϕ) be the standard K-template for γ, and define the K′-template (T′, f, ϕ′) as in the paragraph preceding the statement of the Theorem. We may assume, after moving x and y a uniformly bounded distance if necessary (see part 3 of definition 4.4), that x = ϕ(x_{1}), y = ϕ(y_{1}) for some x_{1}, y_{1} ∈ T . We get (6.6) by applying Theorem 5.1 twice – once to (T , f, ϕ) and once to (T′, f, ϕ′). Specifically, Since z ∈ \overline{ϕ(x_{1})ϕ(y_{1})} an application of Theorem 5.1 part 2 gives

d_{X}(z, ϕ(\overline{x_{1}y_{1}})) ≤ (1 + d_{X}(z, x))θ_{(X,K)}(d_{X}(z, x)).

So there is a z_{1} ∈ \overline{x_{1}y_{1}} with d_{X}(z, ϕ(z_{1})) ≤ (1 + d_{X}(z, x))θ_{(X,K)}(d_{X}(z, x)). Hence we see

d_{X′}(Φ(z), ϕ′(z_{1})) ≤ L(1 + d_{X}(z, x))θ_{(X,K)}(d_{X}(z, x)) + A.

Now an application of Theorem 5.1 part 1 to ϕ′ gives

d_{X′}(ϕ′(z_{1}), \overline{Φ(x)Φ(y)}) = d_{X′}(ϕ′(z_{1}), \overline{ϕ′(x_{1})ϕ′(y_{1})}) ≤ (1+d_{T′}(x_{1}, z_{1}))θ_{(X′,K′)}(d_{T′}(x_{1}, z_{1})).

We thus need to bound d_{T′}(x_{1}, z_{1}) linearly from above by d_{X}(z, x). But this follows since d_{T′}(x_{1}, z_{1}) ≤ L′d_{X′}(ϕ′(x_{1}), ϕ′(z_{1})) + A′ (where L′ and A′ come from Proposition 4.6 and depend only on K′ and the geometry of X′), d_{X′}(ϕ′(x_{1}), ϕ′(z_{1})) ≤ Ld_{X}(ϕ(x_{1}), ϕ(z_{1})) + A, and d_{X}(ϕ(x_{1}), ϕ(z_{1})) ≤ d_{X}(x, z) + d_{X}(z, ϕ(z_{1})) ≤ d_{X}(x, z) + (1 + d_{X}(z, x))θ_{(X,K)}(d_{X}(z, x)). Equation (6.6) will thus follow for an appropriate choice θ_{(K,L,A,X,X′)} that will depend only on L, A, K, and the geometry of X and X′. Lemma 2.5 then applies to Φ, so we get an induced embedding ∂_{∞}Φ : ∂_{∞}X → ∂_{∞}X′ which is automatically G-equivariant.

Now suppose λ = μ = a. Pick η ∈ ∂_{∞}T, and a geodesic ray γ ⊂ T with ∂_{∞}γ = η. Then the identity map T → T′ between the associated templates (T , f, ϕ) and (T′, f, ϕ′) constructed as above is a homothety, and so part 5 of Theorem 5.1 applied to ∂_{∞}ϕ|_{∂_{∞}^{∞}T} and ∂_{∞}ϕ′|_{∂_{∞}^{∞}T′} then shows that ∂_{∞}Φ|_{∂_{∞}^{η}X} induces an isometry ∂_{T}^{η} X → ∂_{T}^{η} X′. By Corollary 5.29 it remains only to show that ∂_{∞}Φ induces an isometry ∂_{T} X_{v} → ∂_{T} X′_{v}. But Lemma 6.4 part 2 implies that Φ_{v} : Y_{v} → Y′_{v} is at finite distance from a product of Hausdorff approximations (up to rescaling of X′ by \frac{1}{a}) and so Φ_{v} induces an isometry ∂_{T} X_{v} → ∂_{T} X′_{v}.

7. Recovering the geometric data from the action on the ideal boundary

In this section we will prove the remaining implication of Theorem 1.3: for any admissible action G ↷ X, the topological conjugacy class of the action G ↷ ∂_{∞}X determines the functions MLS_{v} : G_{v} → ℝ_{+} and τ_{v} : G_{v} → ℝ up to a multiplicative factor, for every vertex v ∈ G. Our strategy for proving this is as follows. Using Lemma 3.23 and Corollary 5.26, for any ideal boundary point η of the Bass-Serre tree T, we may detect the subset ∂_{∞}^{η}X ⊂ ∂_{∞}X; specifically, the action G ↷ ∂_{∞}X determines the set of boundary points η ∈ ∂_{∞}T for which |∂_{∞}^{η}X| = 1. To extract useful information from this, we consider a special class of geodesic rays γ ⊂ T (see Definition 7.11) which admit templates (T , f, ϕ) where T is asymptotically selfsimilar: there is a (non-surjective) map T → T which stretches distances by a factor of 2. These templates have two key properties: their geometry relates directly to the geometric data MLS_{v} and τ_{v}, and at the same time we can tell explicitly when (in terms of the geometry) we have |∂_{∞}^{∂_{∞}γ}T | = 1 (see section 7.2). Putting all this together we able to recover the geometric data from the action G ↷ ∂_{∞}X.

7.1. ∂_{T}^{∞}T is either a point or an interval

Lemma 7.1. Let X be a Hadamard space, p ∈ X, and γ_{i} ⊂ X a sequence of geodesics with lim_{i→∞} d(p, γ_{i}) = ∞. Set

S := \{ξ ∈ ∂_{T} X | \overline{pξ} ∩ γ_{i} ≠ ∅ for all i\}.

Then S embeds isometrically in the interval [0, π].

Proof. Pick x, y, z ∈ S, and for each i choose x_{i} ∈ \overline{px} ∩ γ_{i}, y_{i} ∈ \overline{py} ∩ γ_{i}, z_{i} ∈ \overline{pz} ∩ γ_{i}. After passing to a subsequence and reordering x, y, z we may assume that y_{i} lies between x_{i} and z_{i} on γ_{i}, or that it coincides with x_{i} or z_{i}. Then it follows that

˜∠_{p}(x_{i}, y_{i}) + ˜∠_{p}(y_{i}, z_{i}) ≤ ˜∠_{p}(x_{i}, z_{i}).

Taking the limit as i → ∞, we get

∠_{T} (x, y) + ∠_{T} (y, z) ≤ ∠_{T} (x, z).

Hence ({x, y, z}, ∠_{T}) ⊂ ∂_{T} X embeds isometrically in [0, π].

If |S| = 1 the lemma is immediate, so assume |S| ≥ 2, and construct a map f : S → ℝ as follows. Pick distinct points x_{0}, x_{1} ∈ S and for i = 0, 1 choose f(x_{i}) ∈ ℝ so that d(x_{0}, x_{1}) = d(f(x_{0}), f(x_{1})). Now define f uniquely by the condition that d(f(s), f(x_{i})) = d(s, x_{i}) for all s ∈ S and i = 0, 1. Clearly f is an isometric embedding, and its image lies in an interval of length at most π.

Lemma 7.2. Let T be a half template with walls W_{0}, W_{1}, . . .. For i ≥ 1 set α_{i} := α(W_{i}) where α : W all_{T}^{0} → (0, π) is the angle function, and assume min{α_{i}, π − α_{i}} ≥ β for all i ≥ 1. Then

1. The diameter of ∂_{T}^{∞}T with respect to the Tits metric is at most π − β.

2. For p ∈ W_{0} and every i > 0 there is an R_{i}, depending only on β and d(o_{i}, p), so that if q ∈ L_{i}^{+} then \overline{pq} ∩ L_{i}^{−} ⊂ B(o_{i}, R_{i}).

[figure: o_{i}; y_{i}^{−}; ≤ π; x_{i}^{−}; ≤ π-β; y_{i}^{+}; x_{i}^{+}; L_{i}^{−}; L_{i}^{+}; ≥ β]

Figure 2: On the wall W_{i} the angle between L_{i}^{−} and L_{i}^{+} is ≥ β hence the angle at x_{i}^{−} is ≤ π − β. The angle at y_{i}^{−} is always ≤ π (no mater where y_{i}^{+} lies on L_{i}^{+}).

Proof. Pick p ∈ W_{0} ⊂ T , and distinct points x, y ∈ ∂_{T}^{∞}T . Let x_{i}^{−} ∈ L_{i}^{−} (resp. x_{i}^{+} ∈ L_{i}^{+}) be the point where the ray \overline{px} enters (resp. exits) the wall W_{i}; define y_{i}^{−}, y_{i}^{+} similarly. Since x ≠ y, there is an i_{0} > 0 so that x_{i}^{−} ≠ y_{i}^{−} when i ≥ i_{0}. Pick i ≥ i_{0}, and assume that d(x_{i}^{−}, o_{i}) ≥ d(y_{i}^{−}, o_{i}) (the case when d(y_{i}^{−}, o_{i}) ≥ d(x_{i}^{−}, o_{i}) is similar). Clearly (see figure 2) we have

∠_{x_{i}^{−}}(x_{i}^{+}, o_{i}) = ∠_{x_{i}^{−}}(x_{i}^{+}, y_{i}^{−}) ≤ max{α_{i}, π − α_{i}} ≤ π − β.

Now this implies 1 since by standard properties of Tits angles

∠_{T} (x, y) = [ lim_{i→∞}(∠_{x_{i}^{−}}(y_{i}^{−}, x) + ∠_{y_{i}^{−}}(x_{i}^{−}, y))] − π ≤ max{α_{i}, π − α_{i}} ≤ π − β.

It also implies 2 by applying triangle comparison to the angle at x_{i} of the triangle with vertices p, o_{i}, and x_{i}.

Proposition 7.3. Let T be a uniform half template with min{α_{i}, π − α_{i}} ≥ β for all i. Then ∂_{T}^{∞}T is isometric to an interval [0, θ] where θ ∈ [0, β].

Proof. Applying Lemma 7.1 with γ_{i} = L_{i}^{−} and Lemma 7.2 we get that ∂_{T}^{∞}T is isometric to a subset of [0, β]. We need only show that ∂_{T}^{∞}T is connected. Suppose x, y ∈ ∂_{T}^{∞}T , and pick a point z lying on the Tits interval \overline{xy} ⊂ ∂_{T} T . Let p, x_{i}^{−}, y_{i}^{−} be as in the proof of Lemma 7.2. Then lim_{i→∞} ˜∠_{p}(x_{i}^{−}, y_{i}^{−}) = ∠_{T} (x, y), so we can choose a sequence z_{i} ∈ \overline{x_{i}^{−}y_{i}^{−}} ⊂ L_{i}^{−} so that lim_{i→∞} ˜∠_{p}(x_{i}^{−}, z_{i}) = ∠_{T} (x, z) and lim_{i→∞} ˜∠_{p}(z_{i}^{−}, y_{i}) = ∠_{T} (z, y). The segments \overline{pz_{i}} converge to the ray \overline{pz}. Since for any j > 0 the segment \overline{pz_{i}} crosses L_{j}^{+} for sufficiently large i, by Lemma 7.2 part 2 we get

\overline{pz_{i}} ∩ L_{j}^{−} ⊂ B(o_{j}, R_{j})

and we conclude that \overline{pz} ∩ L_{j}^{−} ≠ ∅. Hence z ∈ ∂_{T}^{∞}T .

The θ in the above proposition is referred to as the Tits angle of T and is denoted θ(T ).

7.2. Self-similar Templates

In this section we study a special class of full templates called self-similar templates.

Definition 7.4. Let T be a full template with W all_{T} = {W_{i}}_{i∈ℤ} and Strip_{T} = {S_{i}}_{i∈ℤ}, and set α_{i} := α(W_{i}), l_{i} := l(S_{i}), and ϵ_{i} := ϵ(S_{i}) (we define ϵ using the strip orientation compatible with the strip directions and the usual ordering on ℤ). Then T is a self-similar template if for all i, j ∈ ℤ we have α_{i} = α_{j}, l_{i+2j} = 2^{j}l_{i}, and ϵ_{i+2j} = 2^{j}ϵ_{i}. In this case we say that T has data (β; l_{0}, ϵ_{0}, l_{1}, ϵ_{1}) where β = α_{i} for all i ∈ ℤ.

Note that by the definition, if we rescale the metric on a self-similar template T by a factor of 2, then we get a template equivalent to T , i.e. there is a homothety Φ : T → T which preserves strip directions, stretches distances by a factor of 2, and which shifts wall and strip indices by 2. A self-similar template is determined up to equivalence by β and the data {l_{0}, ϵ_{0}, l_{1}, ϵ_{1}}.

T contains one point v which is not on any wall or strip (since the union of the walls and strips is not complete). The point v is the limit of the Cauchy sequence {o_{−i}|i = 0, 1, 2, ...}. On the other hand, for each j ∈ ℤ the half template, T_{j}, given by the union of planes W_{i} and strips S_{i} for i ≥ j is uniform (and complete). The images of the embeddings ∂_{∞}^{∞}T_{i} → ∂_{∞}T and ∂_{T}^{∞}T_{i} → ∂_{T} T are independent of i, and we use ∂_{∞}^{∞}T (respectively ∂_{T}^{∞}T ) to denote this common subspace. We will say that T is trivial if |∂_{∞}^{∞}T | = 1 and nontrivial if |∂_{∞}^{∞}T | > 1.

We note that if r is a geodesic ray parameterized by arclength then Φ ∘ r is a geodesic ray parameterized by twice arclength. So Φ and Φ^{−1} take rays to rays. Since Φ is a homothety it preserves the Tits angles between rays.

If R_{i} represents the space of geodesic rays starting at o_{i} and intersecting W_{j} for all j ≥ i, then the above shows Φ^{−1}(R_{i}) = R_{i−2} and that Φ^{−1} acts as a Tits isometry on ∂_{T}^{∞}T . In particular since ∂_{T}^{∞}T is isometric to an interval, Φ leaves the midpoint fixed, and Φ^{2} acts as the identity. Thus by repeated applications of Φ^{−1} we see that we can represent ∂_{T}^{∞}T as the set rays, R_{−∞}, that start at v and are invariant under Φ^{2}. Further, the middle ray is invariant under Φ.

We will show in the lemma below that all rays in R_{−∞} are Φ invariant. In particular any such ray that intersects an o_{i} must intersect all o_{i+2n} for n ∈ ℤ, and hence there are at most two such rays, r_{even} and r_{odd}, in R_{−∞}.

Each choice N ∈ {I, II, III, IV } determines a choice of quarter planes Q_{i}^{N} ⊂ W_{i} as follows: for all k ∈ ℤ, Q_{2k}^{N} = Q_{N} ⊂ W_{2k} while Q_{2k+1}^{N} = −Q_{N} ⊂ W_{2k+1}. Straightforward Euclidean geometry shows that the corresponding development map, D_{N}, with D_{N}(v) = 0 has the property that D_{N} ∘ Φ ∘ D_{N}^{−1} is multiplication by 2 wherever (and however) it is defined (see Figure 3). (The easiest way to see this is to first develop Q_{0}^{N}, S_{0}, Q_{1}^{N}, S_{1}, and Q_{2}^{N} into the plane, then shift the origin (0, 0) so that it lies on the line through D(o_{0}) and D(o_{2}) and such that D(o_{0}) lies between (0, 0) and D(o_{2}) and such that the distance from (0, 0) to D(o_{2}) is twice the distance to D(o_{0}). We note that D(Q_{2}^{N}) = 2D(Q_{0}^{N}). We can now define a map D uniquely so that D ∘ Φ ∘ D^{−1} is multiplication by 2. It is easy to check that this map is a development and hence by uniqueness is D_{N} - up to an element of O(2).) There are rays r_{even} and r_{odd} from the origin such that D_{N}(o_{2i}) ∈ r_{even} and D_{N}(o_{2i+2}) ∈ r_{odd}. (We can fix D_{N} completely if desired by taking r_{even} to be the positive x-axis and to make r_{odd} point in the upper half plane.)

Lemma 7.5. Let T be a nontrivial self-similar template. Then there is a choice

[figure: o_{0}; I; o_{2}; I; o_{4}; I; o_{6}; I; I; o_{1}; III; o_{3}; III; o_{5}; III; o_{7}; III; S_{6}; S_{5}; r_{even}; r_{odd}]

Figure 3: Development of a self-similar template from quarter planes Q_{i}^{I}. Developements of quadrants for nonnegative integers are shaded and labeled with their type, while those with negative indices are not shown. The direction of S_{5} and S_{6} are shown via arrows. The interval of rays between r_{even} and r_{odd} correspond to rays in the template.

N ∈ {I, II, III, IV } such that for r ∈ R_{−∞}, r ∩ W_{i} ⊂ Q_{i}^{N}. Each r ∈ R_{−∞} is Φ invariant, and r_{even} and r_{odd} are the D_{N} images of the boundary rays of R_{−∞} as a closed interval. In particular the set of r ∈ R_{−∞} span a space isometric, via D_{N} to a convex cone in R^{2}.

Proof. The fact that R_{−∞} is a closed interval follows from Proposition 7.3. If r_{1}, r_{2} ∈ R_{−∞} then their invariance under the homothety Φ^{2} implies that ∠_{T} (r_{1}, r_{2}) = ∠_{v}(r_{1}, r_{2}), and hence the rays between r_{1} and r_{2} span a space isometric to a convex cone in R^{2}. In particular, no ray between r_{1} and r_{2} can intersect an origin o_{i}. Thus the middle (Φ invariant) ray r_{m} determines N, M ∈ {I, II, III, IV } such that r_{m} ∩ W_{2n} ⊂ Q_{N}, and r_{m} ∩ W_{2n+1} ⊂ Q_{M}. The above says that for any interior ray, r, r ∩ W_{2n} ⊂ Q_{N}, and r ∩ W_{2n+1} ⊂ Q_{M}. Consider the development map D determined by this choice of quarter planes. We will assume that D(v) is the origin, so in particular D(Q_{i}) and D(S_{i}) miss the origin. Now D ∘ Φ^{2} ∘ D^{−1} is defined when restricted to the image of each strip or quarter plane. It is clearly just multiplication by 4, since that is what it does to each ray in the D image of R_{−∞}.

We assume that N = I (the other cases are similar) and must prove that M = III. If M = II (resp. M = IV) then for i = 1 (resp. i = 0), D(Q_{i}) ∪ D(S_{i}) ∪ D(Q_{i+1}) will contain a half plane which in turn contains one of D(Q_{i}) or D(Q_{i+1}). But this cannot happen since any ray entering that half plane will never leave it. If M = I then for k ≥ max\{\frac{π}{β}, \frac{π}{π−β} \} we have ∪_{i=1}^{k}(D(Q_{i}) ∪ DS_{i}) contains a half plane which in turn contains one of the quarter planes and the same argument works. Thus we conclude that M = III, and Q_{i} = Q_{i}^{N}.

We know that D_{N} ∘ Φ ∘ D_{N}^{−1} is just multiplication by 2. This means in particular that all rays in R_{−∞} are invariant under Φ. Now since at least one ray between r_{even}

[figure: -\frac{π}{2}; \frac{π}{2}; β-\frac{π}{2}; \frac{π}{2}; -\frac{π}{2}]

Figure 4: The graph of A_{β} for β = \frac{3π}{5}.

and r_{odd} hits every D_{N}(Q_{i}) this is true of all such rays, hence all are D_{N} of a ray in T . This gives a 1-1 correspondence between rays in T and rays in the plane between r_{even} and r_{odd} completing the lemma.

The argument in the last proof shows that a self-similar template is nontrivial if and only if there is an N ∈ {I, II, III, IV } such that some (and hence any) ray between the corresponding r_{even} and r_{odd} intersects every D_{N}(Q_{i}^{N}). But by selfsimilarity this will be true if and only if the line segment from D_{N}(o_{0}) to D_{N}(o_{2}) intersects D_{N}(Q_{1}^{N}). This leads to the following lemma:

We will use the notation ℝ_{0}^{4} = {(x_{1}, x_{2}, x_{3}, x_{4}) ⊂ ℝ^{4} | x_{1}, x_{3} > 0}.

Let A_{β} be:

\{(x, y) ∈ (−\frac{π}{2} , \frac{π}{2} )×(−\frac{π}{2} , \frac{π}{2} ) | x + β ≥ y ≥ x − β and −x + (π − β) ≥ y ≥ −x − (π − β)\}.

See Figure 4.

Lemma 7.6. Fix π > β > 0 and let A_{β} be the set of (l_{0}, ϵ_{0}, l_{1}, ϵ_{1}) ⊂ ℝ_{0}^{4} such that the self-similar template with data (β; l_{0}, ϵ_{0}, l_{1}, ϵ_{1}) is trivial. Then

A_{β} = \{(l_{0}, ϵ_{0}, l_{1}, ϵ_{1}) ∈ ℝ_{0}^{4} | (arctan(\frac{ϵ_{0}}{l_{0}} ), arctan(\frac{ϵ_{1}}{l_{1}} )) ∈ A_{β}\}

Proof. Let ψ_{i} = arctan(\frac{ϵ_{i}}{l_{i}}). The proof follows once we show that R^{4} − A_{β} consists of 4 components given in order for N=I, II, III, and IV by:

ψ_{0} > ψ_{1} + β, −ψ_{1} − (π − β) > ψ_{0}, ψ_{1} − β > ψ_{0}, and ψ_{0} > −ψ_{1} + (π − β).

Here we do the case N = I, i.e. the line segment from D_{I}(o_{0}) to D_{I}(o_{2}) intersects D_{I}(Q_{1}^{I}) . The cases N = II, III, and IV are similar.

Let π > θ_{0} > 0 be the angle between the line segment from D_{I}(o_{1}) to D_{I}(o_{0}) and the “incoming” edge of D_{I}(Q_{1}^{I}) (see Figure 5). Here the “incoming” edge is the D_{I} image of the negative half line of L_{1}^{−} (since Q_{1}^{I} is of type III). Similarly let θ_{2} be the angle between the line segment from D_{I}(o_{1}) to D_{I}(o_{2}) and the “outgoing” edge of D_{I}(Q_{1}^{I}). θ_{1} will be the angle at D_{I}(o_{1}) of the sector D_{I}(Q_{1}^{I}) (which in our case is just β). It is easy to see that our condition is equivalent to

π > θ_{0} + θ_{1} + θ_{2}.

[figure: I; I; III; l_{0}; l_{1}; ϵ_{0}; ϵ_{1}; θ_{0}; θ_{1}; θ_{2}; β; β]

Figure 5: Developement in case I; ϵ_{0} > 0, ϵ_{1} < 0 .

The rest of the argument is just that the definitions (being careful about how the sign of ϵ_{i} and the orientation of the strips interact) give us, when N = I,

θ_{0} = \frac{π}{2} − arctan(\frac{ϵ_{0}}{l_{0}} ) and θ_{2} = \frac{π}{2} − arctan(\frac{−ϵ_{1}}{l_{1}} )

which completes the argument.

We extract the information we need with the following elementary (but somewhat non-trivial):

Corollary 7.7. Pick a_{i} > 0 for i = 1, …, 4, b_{1} > 0, b_{2} ∈ ℝ, and a subset B ⊂ ℝ_{0}^{4}. Suppose there is a β ∈ (0, π) so that B is precisely the set of (x_{1}, x_{2}, x_{3}, x_{4}) ∈ ℝ_{0}^{4} for which

(a_{1}x_{1} + b_{1}, a_{2}x_{2} + b_{2}, a_{3}x_{3}, a_{4}x_{4}) ∈ A_{β}.

Then β, \frac{b_{1}}{a_{1}} , and \frac{b_{2}}{a_{2}} are uniquely determined by B. If this unique β is not \frac{π}{2} then \frac{b_{1}}{a_{2}} and \frac{b_{2}}{a_{1}} are also determined by B.

Proof. We consider the map

Ψ(x_{1}, x_{2}, x_{3}, x_{4}) = (arctan(\frac{a_{2}x_{2} + b_{2}}{a_{1}x_{1} + b_{1}} ), arctan(\frac{a_{4}x_{4}}{a_{3}x_{3}} )) =: (ψ_{0}(x_{1}, x_{2}), ψ_{1}(x_{3}, x_{4})).

the previous lemma along with our assumption says that for a_{1}x_{1} +b_{1} > 0 and x_{3} > 0 (x_{1}, x_{2}, x_{3}, x_{4}) ∈ B ⟺ Ψ(x_{1}, x_{2}, x_{3}, x_{4}) ∈ A_{β}. Ψ maps {(x_{1}, x_{2}, x_{3}, x_{4})|a_{1}x_{1} + b_{1} > 0 and x_{3} > 0\} onto (−\frac{π}{2}, \frac{π}{2}) × (−\frac{π}{2}, \frac{π}{2}).

Step 1: β is determined by B. We notice that (x, y^{0}) ∈ A_{β} for all \frac{π}{2} − ϵ < x < \frac{π}{2} if and only if y^{0} = \frac{π}{2} − β. Similarly (x^{1}, y) ∈ A for all large \frac{π}{2} − ϵ < y < \frac{π}{2} if and only if x^{1} = \frac{π}{2} − β. Thus (1, x_{2}, x_{3}^{0}, x_{4}^{0}) ∈ B for all sufficiently large x_{2} if and only if ψ_{1}((x_{3}^{0}, x_{4}^{0})) = \frac{π}{2} − β, while (x_{1}^{1}, x_{2}^{1}, 1, x_{4}) ∈ B for all sufficiently large x_{4} if and only if ψ_{0}((x_{1}^{1}, x_{2}^{1})) = \frac{π}{2} − β. Fix such an (x_{1}^{1}, x_{2}^{1}), then, if β ≠ \frac{π}{2}, there is a unique x_{4}^{1} such that for all x_{4} ≥ x_{4}^{1}, (x_{1}^{1}, x_{2}^{1}, 1, x_{4}) ∈ B. If β < \frac{π}{2} we have ψ_{1}(1, x_{4}^{1}) = \frac{π}{2} − 2β, while if β > \frac{π}{2} we have ψ_{1}(1, x_{4}^{1}) = 2β − \frac{3π}{2}.

Let ā_{i}, b̄_{1}, b̄_{2}, and β̄ be another choice of parameters that work and Ψ̄ and ψ̄_{i} the corresponding functions. Then tan(ψ̄_{1}) = c tan(ψ_{1}) where c = \frac{ā_{4}a_{3}}{ā_{3}a_{4}} > 0. Plugging in (x_{3}^{0}, x_{4}^{0}) from the previous paragraph allows us to conclude tan(\frac{π}{2} − β̄) = c tan(\frac{π}{2} − β), (i.e. tan(β̄) = \frac{1}{c} tan(β) if β ≠ \frac{π}{2}). Note in particular that the sign of \frac{π}{2} − β̄ is the same as the sign of \frac{π}{2} − β. Plugging in (1, x_{4}^{1}) from the above paragraph we get tan(\frac{π}{2} − 2β̄) = c tan(\frac{π}{2} − 2β) if β < \frac{π}{2} and tan(2β̄ − \frac{3π}{2}) = c tan(2β − \frac{3π}{2}) if β > \frac{π}{2}. However this can only happen if c=1 and β = β̄. Thus we conclude that β is determined by B and that either β = \frac{π}{2} or else c = 1 and ψ_{1} = ψ̄_{1}.

Step 2: If β ≠ \frac{π}{2} then ψ_{0} = ψ̄_{0}, while if β = \frac{π}{2} then tan(ψ̄_{0}) = \frac{1}{c} tan(ψ_{0}). We will first show that given B, β and ψ_{1} then there is at most one choice for ψ_{0}. To see this fix (x_{1}, x_{2}) and consider S ⊂ R^{2} such that B ∩ ((x_{1}, x_{2}) × R^{2}) = (x_{1}, x_{2}) × S. Then ψ_{1}(S) is precisely the interval such that A_{β} ∩ (ψ_{0}(x_{1}, x_{2}) × R) = (ψ_{0}(x_{1}, x_{2}) × ψ_{1}(S)). However the shape of A_{β} is such that ψ_{1}(S) thus determines ψ_{0}(x_{1}, x_{2}), unless β = \frac{π}{2} in which case it determines ψ_{0}(x_{1}, x_{2}) up to sign. The sign is determined by continuity and the fact that for x_{2} large and positive (resp. negative) then ψ_{0}(x_{1}, x_{2}) is positive (resp. negative).

Thus if β ≠ \frac{π}{2} then ψ_{1} = ψ̄_{1} and we see that ψ̄_{0} = ψ_{0} is the unique solution. If β = \frac{π}{2} then tan(ψ̄_{1}) = c tan(ψ_{1}) and A_{β} = \{(x, y) ∈ R^{2}| − x + \frac{π}{2} ≥ y ≥ x − \frac{π}{2} when x > 0 and x + \frac{π}{2} ≥ y ≥ −x − \frac{π}{2} when x < 0\}. Thus for c > 0 the map (x, y) → (arctan(\frac{1}{c} tan(x)), arctan(c tan(y)) preserves A_{β} and step 2 follows.

The rest of the proof follows from the following equation that holds for all positive x_{1} and all x_{2}:

\frac{ā_{2}x_{2} + b̄_{2}}{ā_{1}x_{1} + b̄_{1}} = \frac{1}{c} \frac{a_{2}x_{2} + b_{2}}{a_{1}x_{1} + b_{1}}

and the fact that c = 1 when β ≠ \frac{π}{2}.

7.3. Recovering the data

In this section G will denote a fixed admissible graph of groups, G := π_{1}(G) the fundamental group of G, and G ↷ T the Bass-Serre action for G. For every vertex v of G we choose a generator ζ_{v} ∈ Z(G_{v}) for the center of Z(G_{v}) as in Definition 3.9. Also, we will fix an admissible action G ↷ X. We will use the template notation from section 4.1. Recall that when T is a half-template, then ∂_{∞}^{∞}T ⊂ ∂_{∞}T denotes the set of boundary points corresponding to rays which intersect all but finitely many walls.

The goal of this section is to prove the remaining half of Theorem 1.3: the topological conjugacy class of the action G ↷ ∂_{∞}X determines the functions MLS_{v} : G_{v} → ℝ_{+} and τ_{v} : G_{v} → ℝ up to a multiplicative factor, for every vertex v ∈ G.

Definition 7.8. An element g of a vertex group G_{v} is restricted if g acts on Ȳ_{v} (see section 3.2) as an axial isometry and its fixed points in ∂_{∞}Ȳ_{v} are distinct from the fixed points of G_{e} where e is any edge incident to v.

It will be convenient to choose, for each vertex v of G, a restricted element δ_{v} ∈ G_{v} which lies in the commutator subgroup [G_{v}, G_{v}] ⊂ G_{v}:

Lemma 7.9. For every vertex v, the commutator subgroup [G_{v}, G_{v}] contains restricted elements.

Proof. We first recall that H_{v} := G_{v}/Z(G_{v}) is a nonelementary hyperbolic group, and the induced action H_{v} ↷ Ȳ_{v} is discrete and cocompact.

Choose a free nonabelian subgroup S ⊂ H_{v} [Gro87, p. 212], and elements ḡ_{1}, ḡ_{2} ∈ S which belong to a free basis for the commutator subgroup [S, S] ⊂ [H_{v}, H_{v}], and let g_{i} ∈ G_{v} be lift of ḡ_{i} under the projection G_{v} → H_{v}. Then g_{i} acts axially on Ȳ_{v} since ḡ_{i} has infinite order in H_{v} and H_{v} acts discretely on Ȳ_{v}. Note that F ix(g_{1}, ∂_{∞}Ȳ_{v}) ∩ F ix(g_{2}, ∂_{∞}Ȳ_{v}) = ∅, since otherwise by Lemma 2.21 we would have F ix(g_{1}, ∂_{∞}Ȳ_{v}) = F ix(g_{2}, ∂_{∞}Ȳ_{v}), forcing ⟨g_{1}, g_{2}⟩ to be virtually cyclic, which is absurd. Set h_{n} := g_{1}^{n}g_{2}^{n}. Lemma 2.10 tells us that h_{n} is axial for large n and F ix(h_{n}, ∂_{∞}Ȳ_{v}) converges to {ξ_{1}, ξ_{2}} ⊂ ∂_{∞}Ȳ_{v} where ξ_{i} ∈ F ix(g_{i}, ∂_{∞}Ȳ_{v}). The induced action of G_{e} on Ȳ_{v} translates a geodesic γ_{e} ⊂ Ȳ_{v}. By the finiteness of G we can choose elements g_{e} ∈ G_{e} such that the induced translation of g_{e} is nonzero but uniformly bounded. Thus Lemma 2.21 says that subsets F ix(G_{e}, ∂_{∞}Ȳ_{v}) = F ix(g_{e}, ∂_{∞}Ȳ_{v}) define a discrete subset of (∂_{∞}Ȳ_{v} × ∂_{∞}Ȳ_{v})/ℤ_{2}, so either

a) F ix(h_{n}, ∂_{∞}Ȳ_{v})∩F ix(G_{e}, ∂_{∞}Ȳ_{v}) = ∅ for all edges e incident to v when n is large, or

b) There is a subsequence h_{n_{i}} and an edge e incident to v so that F ix(h_{n_{i}}, ∂_{∞}Ȳ_{v}) = F ix(G_{e}, ∂_{∞}Ȳ_{v}).

But if b) held then we would have F ix(G_{e}, ∂_{∞}Ȳ_{v}) = {ξ_{1}, ξ_{2}}, which, by Lemma 2.21, would force the absurd conclusion that F ix(g_{1}, ∂_{∞}Ȳ_{v}) = F ix(g_{e}, ∂_{∞}Ȳ_{v}) = F ix(g_{2}, ∂_{∞}Ȳ_{v}). Hence case a) holds and the lemma is proved.

Notice that for every v, MLS_{v}(δ_{v}) ≠ 0 (since δ_{v} acts on Ȳ_{v} as an axial isometry), τ_{v}(δ_{v}) = 0 since δ_{v} ∈ [G_{v}, G_{v}], and τ_{v}(ζ_{v}) ≠ 0.

Lemma 7.10. In order to determine MLS_{v} and τ_{v} up to a multiplicative factor, it suffices to determine the ratios

\frac{MLS_{v}(σ)}{MLS_{v}(δ_{v})} and \frac{τ_{v}(σ)}{τ_{v}(ζ_{v})}

for every restricted element σ ∈ G_{v} whose fixed point set in ∂_{∞}Ȳ_{v} is disjoint from F ix(δ_{v}, ∂_{∞}Ȳ_{v}).

Proof. Choose an arbitrary σ ∈ G_{v}.

We first discuss MLS_{v}. Note that MLS_{v}(σ) = 0 iff σ projects to an element of finite order in H_{v}; hence we may assume that σ acts on Ȳ_{v} as an axial isometry. First assume that F ix(σ, ∂_{∞}Ȳ_{v}) = F ix(δ_{v}, ∂_{∞}Ȳ_{v}). Let σ̄, δ̄_{v} ∈ H_{v} be the projections to H_{v}. Then σ̄ and δ̄_{v} generate a virtually cyclic subgroup S because they have a common axis. Hence there is a finite subset {s_{1}, …, s_{k}} ⊂ G_{v} so that for any n we have σ^{n} = s_{i_{n}}δ_{v}^{j_{n}}ζ_{v}^{k_{n}} for suitable i_{n}, j_{n}, k_{n}. Then

MLS_{v}(σ) = lim_{n→∞} \frac{1}{n}MLS_{v}(σ^{n}) = lim_{n→∞} MLS_{v}(δ_{v}^{j_{n}})

so we can recover the ratio above in this case.

Now assume F ix(σ, ∂_{∞}Ȳ_{v}) ∩ F ix(δ_{v}, ∂_{∞}Ȳ_{v}) = ∅. Setting h_{k} := δ_{v}^{k}σ^{k^{2}}, we argue as in the proof of Lemma 7.9 to see that for large k, h_{k} is restricted, F ix(h_{k}, ∂_{∞}Ȳ_{v}) ∩ F ix(δ_{v}, ∂_{∞}Ȳ_{v}) = ∅, and |MLS_{v}(h_{k})−kMLS_{v}(δ_{v})−k^{2}MLS_{v}(σ)| is uniformly bounded (by Lemma 2.10). We may then recover the desired ratios from the formula

MLS_{v}(σ) = lim_{k→∞} \frac{1}{k^{2}} MLS_{v}(h_{k}).

We now consider the behavior of τ_{v}. Suppose σ projects to an element of finite order in H_{v}, or F ix(σ, ∂_{∞}Ȳ_{v}) ∩ F ix(δ_{v}, ∂_{∞}Ȳ_{v}) ≠ ∅. In either case we have a finite set {s_{1}, …, s_{k}} ⊂ G_{v} so that σ^{n} = s_{i_{n}}δ_{v}^{j_{n}}ζ_{v}^{k_{n}} for suitable i_{n}, j_{n}, k_{n}. Then

τ_{v}(σ) = lim_{n→∞} \frac{1}{n}τ_{v}(σ^{n}) = lim_{n→∞}[\frac{j_{n}}{n} τ_{v}(δ_{v}) + \frac{k_{n}}{n} τ_{v}(ζ_{v})].

For the case F ix(σ, ∂_{∞}Ȳ_{v}) ∩ F ix(δ_{v}, ∂_{∞}Ȳ_{v}) = ∅, use the same h_{k} as above for large k since τ_{v}(h_{k}) = k^{2}τ(σ).

We now focus our attention on a vertex v̄_{1} of G. Choose an edge ē of G incident to v̄_{1}, and lift ē to an edge e of the Bass-Serre tree. We adopt the notation from section 2.5 for the associated graph of groups G′, G′ := π_{1}(G′), T′ ⊂ T, etc. We fix some restricted element σ ∈ Gv̄_{1} with F ix(σ, ∂_{∞}Ȳ_{v_{1}}) ∩ F ix(δ_{v_{1}}, ∂_{∞}Ȳ_{v_{1}}) = ∅.

Definition 7.11. (Special rays) For every (p, q, r, s) ∈ ℝ_{0}^{4} we define a geodesic ray γ ⊂ T′ and sequences l̂_{i} = l_{i}(p, q, r, s), ϵ̂_{i} = ϵ̂_{i}(p, q, r, s) as follows.

Case 1: ē ⊂ G is embedded. Let e = \overline{v_{1}v_{2}}. For each i ∈ ℕ we set s_{2i−1} := σ^{2^{i−1}}δ_{v_{1}}^{[p2^{i−1}]}ζ_{v_{1}}^{[q2^{i−1}]} and s_{2i} := δ_{v_{2}}^{[r2^{i−1}]}ζ_{v_{2}}^{[s2^{i−1}]}, where [x] denote the integer part of x ∈ ℝ. Then we let γ ⊂ T be the geodesic ray with successive edges

e, s_{1}e, s_{1}s_{2}e, …, s_{1} … s_{k}e, …,  (7.12)

and define, for all i ∈ ℕ, l̂_{2i−1} = 2^{i−1}(MLS_{v_{1}}(σ)+|p|MLS_{v_{1}}(δ_{v_{1}})), l̂_{2i} = 2^{i−1}|r|MLS_{v_{2}}(δ_{v_{2}}); and ϵ̂_{2i+1} := 2^{i}(τ_{v_{1}}(σ) + qτ_{v_{1}}(ζ_{v_{1}})), and ϵ̂_{2i} := 2^{i−1}sτ_{v_{2}}(ζ_{v_{2}}).

Case 2: ē ⊂ G is a loop. Let t be as in section 2.5. For each i ∈ ℕ we set s_{2i−1} := σ^{2^{i−1}}δ_{v_{1}}^{[p2^{i−1}]}ζ_{v_{1}}^{[q2^{i−1}]} and s_{2i} := δ_{v_{1}}^{[r2^{i−1}]}ζ_{v_{1}}^{[s2^{i−1}]}; then we let γ ⊂ T be the geodesic with successive vertices

v_{1}, tv_{1}, ts_{1}t^{−1}v_{1}, ts_{1}t^{−1}s_{2}tv_{1}, … ts_{1}t^{−1}s_{2}t … s_{2k}tv_{1}, …,  (7.13)

and define, for all i ∈ ℕ, l̂_{2i−1} = 2^{i−1}(MLS_{v_{1}}(σ)+|p|MLS_{v_{1}}(δ_{v_{1}})), l̂_{2i} = 2^{i−1}|r|MLS_{v_{1}}(δ_{v_{1}}); and ϵ̂_{2i+1} := 2^{i}(τ_{v_{1}}(σ) + qτ_{v_{1}}(ζ_{v_{1}})), and ϵ̂_{2i} := 2^{i−1}sτ_{v_{1}}(ζ_{v_{1}}).

These rays are useful because they admit templates that are asymptotically selfsimilar:

Lemma 7.14. There is a half template (T , f, ϕ) for γ with walls W all_{T} = {W_{i}}_{i=1}^{∞} and strips Strip_{T} = {S_{i}}_{i=1}^{∞}, so that for i sufficiently large, l(S_{i}) = l̂_{i}, ϵ(S_{i}) = ϵ̂_{i}, and α(W_{i}) = β, where β is the Tits angle between the (positively oriented) ℝ-factors of Y_{v_{1}} and Y_{v_{2}}.

Proof. We will treat the case when ē is embedded; the other case is similar. To prove the lemma we will show that the desired template may be obtained from a standard template for γ by changing the strip widths and strip gluings by a bounded amount. Thus by ”uniformly” we will just mean independent of i, but possibly dependent on all other choices.

For i ≥ 1 we set e_{i} := s_{1} … s_{i−1}e and v_{i} := e_{i} ∩ e_{i+1}. Recall (section 4.2) that the standard template (T , f, ϕ) for γ is constructed using flats F_{e_{i}} ⊂ Y_{e_{i}} and flat strips S_{e_{i},e_{i+1}} ⊂ Y_{v_{i}}, where S_{e_{i},e_{i+1}} = γ_{e_{i},e_{i+1}} × ℝ ⊂ Ȳ_{v_{i}} × ℝ = Y_{v_{i}}. We choose x_{1} ∈ F_{e_{1}} and set x_{i} := s_{1} … s_{i−1}x_{1} ∈ F_{e_{i}}.

Step 1: There is a constant c_{1} so that d(x_{i}, S_{e_{i},e_{i+1}}) < c_{1}, d(x_{i+1}, S_{e_{i},e_{i+1}}) < c_{1} and |W idth(S_{e_{i},e_{i+1}}) − l̂_{i}| < c_{1}. We will do the case when i = 2j − 1 is odd; the even case is similar. Let π : Y_{v_{1}} = Ȳ_{v_{1}} × ℝ → Ȳ_{v_{1}} be the projection map, and set x̄_{1} := π(x_{1}), F̄_{e} := π(F_{e}), F̄_{s_{i}e} := π(s̄_{i}F_{e}) = π(F_{s_{i}e}), and γ_{e,s_{i}e} := π(S_{e,s_{i}e}). We apply (s_{1} … s_{i−1})^{−1} and then π to everything, and are thereby reduced to showing that there is a c_{1} so that d(x̄_{1}, γ_{e,s_{i}e}) < c_{1}, d(s_{i}x̄_{1}, γ_{e,s_{i}e}) < c_{1}, and |d(s_{i}x̄_{1}, x̄_{1}) − l̂_{i}| < c_{1}.

¿From the definition of s_{i} we have s_{i}x̄_{1} = σ^{2^{j−1}}δ_{v_{1}}^{[p2^{j−1}]}x̄_{1} since ζ_{v_{1}} acts trivially on Ȳ_{v_{1}}. Since σ, δ_{v_{1}} ∈ G_{v_{1}} are restricted elements, the sets ∂_{∞}F̄_{e} ⊂ ∂_{∞}Ȳ_{v_{1}} and ∂_{∞}F̄_{s_{i}e} ⊂ ∂_{∞}Ȳ_{v_{1}} are disjoint from the fixed point sets of σ and δ_{v_{1}}; the latter two sets are disjoint by assumption. Therefore we may apply Lemma 2.10 to conclude that when j is sufficiently large,

a) s_{i} : Ȳ_{v_{1}} → Ȳ_{v_{1}} is an axial isometry with an axis γ_{i} ⊂ Ȳ_{v_{1}} at uniformly bounded distance from x̄_{1} and s_{i}x̄_{1}.

b) |d(s_{i}x̄_{1}, x̄_{1}) − l̂_{i}| is uniformly bounded.

c) The attracting (resp. repelling) fixed point of s_{i} in ∂_{∞}Ȳ_{v_{1}} is close to the attracting fixed point, ξ^{+}, of σ (resp. repelling fixed point, ξ^{−}, of δ_{v_{1}}^{sign(p)}).

Let γ_{e,s_{i}e} be the shortest path from F̄_{e} to F̄_{s_{i}e} with endpoints z̄_{i} ∈ F̄_{e} and w̄_{i} ∈ F̄_{s_{i}e} Let γ be a geodesic with endpoints ξ^{−} and ξ^{+}. The Gromov hyperbolicity of Ȳ_{v_{1}} and Lemma 2.9 part 3 imply that for t fixed and large and i large the axis of s_{i} comes within a uniform distance of γ_{e,s_{i}e}(t) and hence γ_{e,s_{i}e}(t) stay a uniform distance from γ. Thus z̄_{i}, which is the point on F̄_{e} closest to γ_{e,s_{i}e}(t), must stay a uniform distance from the set of points on F̄_{e} closest (in a Buseman function sense) to ξ^{+} (which we see by taking t large) and hence stay uniformly close to x̄_{1} . Similarly s_{i}^{−1}(w̄_{i}) approaches the set of points on F̄_{e} closest to ξ^{−} and hence w̄_{i} stays uniformly close to s_{i}(x̄_{1}).

Step 2: There is a c_{2} so that the standard template (T , f, ϕ) satisfies |ϵ(S_{i}) − ϵ̂_{i}| < c_{2} and α(W_{i}) = β for all i > 1. The assertion that α(W_{i}) = β is clear from the definition of β(W_{i}) and the construction of standard templates. Step 1 then implies that there is a c_{3} so that the origin o_{i} ∈ W_{i} maps under f : T → X to within distance c_{3} of x_{i}, for i > 1. Hence from the definition of ϵ : Strip_{T}^{o} → ℝ we see that ϵ(S_{i}) agrees with τ_{v_{i}}((s_{1} … s_{i−1})s_{i}(s_{1} … s_{i−1})^{−1}) to within 2c_{3}. The conjugacy invariance of τ_{v} then gives |ϵ(S_{i}) − ϵ̂_{i}| < c_{2} for a suitable c_{2}.

Step 3: Adjusting (T , f, ϕ). In steps 1 and 2 we have shown that the standard template satisfies conditions of Lemma 7.14 to within bounded error. So we now modify the construction of T by changing the metric^{11} on Ŝ_{e_{i},e_{i+1}}so that W idth(Ŝ_{e_{i},e_{i+1}}) = l̂_{i} if l_{i} ≥ 1, and leaving Ŝ_{e_{i},e_{i+1}} untouched otherwise, and by modifying the gluings ∂Ŝ_{e_{i},e_{i+1}} → W_{e_{i}} ⨿ W_{e_{i+1}} by a bounded amount so that ϵ(S_{i}) = ϵ̂_{i}. Finally, if we redefine f : T → X to agree with the original f on ⨿_{e} W_{e} and on T − (⨿_{e} W_{e}), then we get the desired template for γ.

Proof the Theorem 1.3 concluded. Consider the subset B ⊂ ℝ_{0}^{4} of 4-tuples (p, q, r, s) for which the geodesic ray γ ⊂ T defined above gives a trivial subset ∂_{∞}^{∂_{∞}γ}X (i.e. a single point); by Lemma 3.23 and Corollary 5.26 the subset ∂_{∞}^{∂_{∞}γ}X can be detected just using the action G ↷ ∂_{∞}X, and so B is also determined by the action G ↷ ∂_{∞}X. On the other hand, by Theorem 5.1, ∂_{∞}^{∂_{∞}γ}X is trivial iff ∂_{∞}^{∞}T is trivial (i.e. a single point) where (T , f, ϕ) is any template for γ. Using Lemma 7.14 we arrive at the following: the subset B of (p, q, r, s) ∈ ℝ_{0}^{4} so that any template T with l(S_{i}) = l̂_{i}(p, q, r, s), ϵ(S_{i}) = ϵ̂_{i}(p, q, r, s), and α(W_{i}) = β (for i sufficiently large) is trivial, is determined by the action G ↷ ∂_{∞}X. But since a template T with l(S_{i}) = l̂_{i}(p, q, r, s), ϵ(S_{i}) = ϵ̂_{i}(p, q, r, s), and α(W_{i}) = β (for i sufficiently large) is trivial iff the self-similar template with data {β; l̂_{3}, ϵ̂_{3}, l̂_{4}, ϵ̂_{4}} is trivial, we may apply Corollary 7.7 to conclude that the ratios

11We do this in the simplest way: we start with the metric product decomposition Ŝ_{e_{i},e_{i+1}} ≃ I ×ℝ and then scale the metric on the I factor.

\frac{MLS_{v_{1}}(σ)}{MLS_{v_{1}}(δ_{v_{1}})} and \frac{τ_{v_{1}}(σ)}{τ_{v_{1}}(ζ_{v_{1}})}

as well as β are determined by the action G ↷ ∂_{∞}X. Moreover, unless β = \frac{π}{2} then

\frac{MLS_{v_{1}}(σ)}{τ_{v_{1}}(ζ_{v_{1}})} and \frac{τ_{v_{1}}(σ)}{MLS_{v_{1}}(δ_{v_{1}})}

are also determined.

8. Examples

In this section we construct the example mentioned in the introduction: we will find two locally compact Hadamard spaces X_{0} and X_{r} on which an admissible group G acts discretely and cocompactly with the same geometric data (i.e. the induced actions of G on ∂_{∞}X_{0} and ∂_{∞}X_{r} are topologically conjugate), an equivariant quasi-isometry F̃ : X_{0} → X_{r}, and a geodesic ray γ of X_{0} such that F̃(γ) does not lie within a bounded distance of a geodesic ray in X_{r}.

To do that we consider for each (small) real r a complex M_{r}, built out of four flat square tori T_{i}. On each T_{i} we use standard angle coordinates (s, t)_{i} with (s + 2nπ, t + 2mπ)_{i} = (s, t)_{i} for any integers n and m. We let M_{r} = T_{1} ∪ T_{2} ∪ T_{3} ∪ T_{4}/ ∼_{r}, where (0, t)_{1} ∼_{r} (t, 0)_{2}, (π, t)_{2} ∼_{r} (t, π)_{3}, (π, t)_{3} ∼ (t, π)_{4} and (0, t)_{4} ∼_{r} (t, r)_{1}. We will let X_{r} represent the universal cover.

The X_{r} are Hadamard spaces with admissible fundamental groups which all have the same geometric data. This is easiest to see by considering M_{r} as the union of four spaces T_{1} ∪ T_{2}, T_{2} ∪ T_{3}, T_{3} ∪ T_{4}, and T_{4} ∪ T_{1} each of which is isometric to a figure eight cross a circle, where both circles in the figure eight and the product circle have length 2π. Each of the three spaces is glued to adjacent spaces along (product) boundary tori (reversing the factors). We note that the underlying finite graph of G is a square with four edges e_{i} corresponding naturally to T_{i}.

It is easy to see that the M_{r} are homeomorphic. In fact for small r the fundamental groups are identified in a natural way. However, we will find it more useful to consider the map F_{r} : M_{0} → M_{r} defined by F (s, t)_{i} = (s, t)_{i}, except that F ((0, t)_{4}) = (t, 0)_{1}. This is not continuous along the closed geodesic (0, t)_{4}, but the induced equivariant quasi-isometry F̃ : X_{0} → X_{r} is relatively easy to study.

We now want to choose a geodesic ray γ in X_{0}. For these spaces geodesics are just geodesics in templates. It is easier to first choose the degenerate half template T that it will lie in. By degenerate template we mean a template where we alow the strip widths to be 0. In fact in our case all the strip widths will be 0, all the angles will be \frac{π}{2}, and the displacements will all be odd multiples of π. We will choose T so that the edges of the ray in the Bass-Serre tree project to the finite graph periodically in the order e_{1}, e_{2}, e_{3}, e_{4}, e_{1} . . .. Subject to this constraint we can still choose the itinerary such that the displacements in T are any odd multiples of π we please. Hence we may choose the itinerary (e.g. make it nontrivial) so that there is a geodesic γ in T which misses all the vertices such that γ ∩ W_{i} gets arbitrarily long. (The easiest way to see this is to consider the developement D. First choose a ray r that you want to be D(γ). The choice of itinerary at each step amounts to a choice of quarter planes among those shifted by 2nπ. We can thus make the choice so that the quarter plane intersects r in increasingly long intervals).

Of course γ is also a geodesic ray in X_{0}. F̃ will take T to a corresponding template T_{r}, and the geodesic ray corresponding to F (γ) must be a geodesic in this template. F̃ will be discontinuous exactly where walls corresponding to e_{4} and e_{1} are glued. The discontinuity is a translation by r perpendicular to the gluing line.

We develop T (and T_{r}) to the plane in such a way that γ goes to a ray with angle \frac{π}{2} > θ > 0, the quarter planes map to planes of type II and IV where walls of type e_{1} and e_{3} yield quarter planes of type II and walls of type e_{2} and e_{4} yield quarter planes of type IV. F̃ will induce a map of developments which is discontinuous precisely on the horizontal lines where the quarter planes coming from e_{4} meet those coming from e_{1}. The discontinuity will be precisely a vertical shift by r. Thus the image of γ consists of arbitrarily long line segments of slope θ with infinitely many vertical jumps of size r. Since the segments get arbitrarily long, the only rays that can stay a bounded distance from F̃(γ) must also have slope θ. But then, because of the infinitely many vertical jumps of size r in F̃(γ), no such ray can stay a bounded distance from F̃(γ).

One can make similar examples on singular piecewise Euclidean graph manifolds. We construct such examples by gluing together two pieces. Each piece is topologically a twice punctured torus cross S^{1}. The boundary of each piece will consist of two totally geodesic square flat two-tori. The first space is constructed by gluing both of the corresponding boundary tori together flipping the coordinates. The second space is similar except that for one of the boundary tori the gluing map is coordinate flipping composed with a small translation.

The metric of each piece is a product metric where the circle has length 1 and where the metric on the torus is the completion of the flat square torus minus two line segments (slits) of length \frac{1}{2}. The torus with the slits is a compact flat singular space with boundary being two closed geodesics of length 1 (i.e. going “around a slit”).

The argument that this gives an example is very similar to the above since geodesics in the space are in fact geodesics in the corresponding templates and since the induced map on the development of appropriate templates has properties similar to the above example.

We also suspect there are such examples on smooth graph manifolds. In fact one may be able to construct such an example by smooth approximations to the above example. To do this carefully it would be necessary to be careful with how closely template geodesics shadow actual geodesics in this case.

References

[Bal95] W. Ballmann. Lectures on spaces of nonpositive curvature. Birkhäuser Verlag, Basel, 1995.

[BS] S. Buyalo and V. Schroeder. On the asymptotic geometry of nonpositively curved graphmanifolds. Trans. Amer. Math. Soc.

[Buy98] S. V. Buyalo. Geodesics in Hadamard spaces. Algebra i Analiz, 10(2):93– 123, 1998.

[CDP90] M. Coornaert, T. Delzant, and A. Papadopoulos. Géométrie et théorie des groupes. Springer-Verlag, Berlin, 1990. Les groupes hyperboliques de Gromov.

[CFF92] C. Croke, A. Fathi, and J. Feldman. The marked length-spectrum of a surface of nonpositive curvature. Topology, 31(4):847–855, 1992.

[CK] C. B. Croke and B. Kleiner. Spaces with nonpositive curvature and their ideal boundaries. To appear in Topology.

[Cro90] C. B. Croke. Rigidity for surfaces of nonpositive curvature. Comment. Math. Helv., 65(1):150–169, 1990.

[DD89] W. Dicks and M. J. Dunwoody. Groups acting on graphs. Cambridge University Press, Cambridge, 1989.

[GdlH90] É. Ghys and P. de la Harpe, editors. Sur les groupes hyperboliques d’après Mikhael Gromov. Birkhäuser Boston Inc., Boston, MA, 1990.

[Gro76] M. Gromov. Three remarks on geodesic dynamics and fundamental groups. Preprint, 1976.

[Gro87] M. Gromov. Hyperbolic groups. In Essays in group theory, pages 75–263. Springer, New York, 1987.

[Gro93] M. Gromov. Asymptotic invariants of infinite groups. In Geometric group theory, Vol. 2 (Sussex, 1991), pages 1–295. Cambridge Univ. Press, Cambridge, 1993.

[HS98] C. Hummel and V. Schroeder. Tits geometry associated with 4-dimensional closed real-analytic manifolds of nonpositive curvature. J. Differential Geom., 48(3):531–555, 1998.

[Lau74] F. Laudenbach. Topologie de la dimension trois: homotopie et isotopie. Société Mathématique de France, Paris, 1974. Astérisque, No. 12.

[Ota90] J. P. Otal. Le spectre marqué des longueurs des surfaces à courbure négative. Ann. of Math. (2), 131(1):151–162, 1990.

[Rua96] K. Ruane. Ph. D Thesis, Florida State University, 1996.

[Sch86] V. Schroeder. Rigidity of nonpositively curved graphmanifolds. Math. Ann., 274(1):19–26, 1986.

[Ser80] J. P. Serre. Trees. Springer-Verlag, Berlin, 1980.

[SW79] P. Scott and T. Wall. Topological methods in group theory. In Homological group theory (Proc. Sympos., Durham, 1977), pages 137–203. Cambridge Univ. Press, Cambridge, 1979.

[Wil] J. Wilson. Preprint.

Chris Croke: Department of Mathematics University of Pennsylvania 209 S. 33rd St. Philadelphia, PA 19104-6395 ccroke@math.upenn.edu

Bruce Kleiner: Department of Mathematics University of Utah Salt Lake City, UT 84112-0090 bkleiner@math.utah.edu Current address: Department of Mathematics University of Michigan Ann Arbor, MI 48109-1109 bkleiner@math.lsa.umich.edu