Spaces with nonpositive curvature and their ideal boundaries

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

February 6, 1998

Abstract

We construct a pair of finite piecewise Euclidean 2-complexes with nonpositive curvature which are homeomorphic but whose universal covers have nonhomeomorphic ideal boundaries, settling a question from [8].

1.1 Introduction

The ideal boundary of a locally compact Hadamard space^{1} X is a compact metrizable space on which the isometry group of X acts by homeomorphisms. Even though the ideal boundary is a well known construct with many applications in the literature (see for example [10, 4, 2]), the action of the isometry group on the boundary has not been studied closely except in the case of symmetric spaces, Gromov hyperbolic spaces, Euclidean buildings, and a handful of other cases. In the Gromov hyperbolic case^{2} the boundary behaves nicely with respect to quasi-isometries: any quasi-isometry f : X_{1} → X_{2} between Gromov hyperbolic Hadamard spaces induces a boundary homeomorphism ∂_{∞}f : ∂_{∞}X_{1} → ∂_{∞}X_{2} [7]. This has the consequence that the ideal boundary is “geometry independent”:

If a finitely generated group G acts discretely, cocompactly and isometrically on two Gromov hyperbolic Hadamard spaces X_{1}, X_{2}, then there is a G-equivariant homeomorphism ∂_{∞}X_{1} → ∂_{∞}X_{2}. In [8, p. 136] Gromov asked whether this fundamental property still holds if the hyperbolicity assumption is dropped. Sergei Buyalo [5] and the authors [6] independently answered Gromov’s question negatively: [5, 6] exhibit a pair of deck group invariant Riemannian metrics on a universal cover which have ideal boundaries homeomorphic to S^{2}, such that the deck group actions on the boundaries are topologically inequivalent. Gromov also asked if ∂_{∞}X_{1} must be (non-equivariantly) homeomorphic to ∂_{∞}X_{2} whenever X_{1} and X_{2} are Hadamard spaces admitting discrete, cocompact, isometric actions by the same finitely generated group G. In this paper we show that even this can fail:

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

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

1Following [3] we will call complete, simply connected length spaces with nonpositive curvature Hadamard spaces.

2The same statement is true of symmetric spaces and Euclidean buildings by [9].

Theorem 1 There is a pair X̄_{1}, X̄_{2} of homeomorphic finite 2-complexes with nonpositive curvature such that the universal covers X_{1}, X_{2} have nonhomeomorphic ideal boundaries.

We remark that if M_{1} and M_{2} are closed Riemannian manifolds with nonpositive curvature and π_{1}(M_{1}) ≃ π_{1}(M_{2}), then their universal covers will have ideal boundaries homeomorphic to spheres of the same dimension.

Although some basic questions about the boundary have now been answered, a number of related issues are wide open, except in a few special cases. It would be interesting to know exactly which geometric features determine the ideal boundary of a Hadamard space up to (equivariant) homeomorphism. This question has a clean answer (see [6]) in the case of graph manifolds or the 2-complexes considered in this paper. In order to answer the question in any generality, it appears that it will be necessary to develop a kind of “generalized symbolic dynamics” for geodesic flows of nonpositively curved spaces.

1.2 Notation and preliminaries

A reference for the facts recalled here is [3]. If X is a Hadamard space, then we denote the ideal boundary of X by ∂_{∞}X, the geodesic segment joining x_{1}, x_{2} ∈ X by \overline{x_{1}x_{2}}, and the geodesic ray leaving p ∈ X in the asymptote class of ξ ∈ ∂_{∞}X by \overline{pξ}. If p ∈ X, ξ_{1}, ξ_{2} ∈ ∂_{∞}X, then ∠_{p}(ξ_{1}, ξ_{2}) is the angle between the initial velocities of the rays \overline{pξ_{1}}, \overline{pξ_{1}}. ∠_{T its}(ξ_{1}, ξ_{2}) := sup_{p∈X} ∠_{p}(ξ_{1}, ξ_{2}) will denote the Tits angle between ξ_{1}, ξ_{2} ∈ ∂_{∞}X. If p ∈ X then ∠_{p}(ξ_{1}, ξ_{2}) = ∠_{T its}(ξ_{1}, ξ_{2}) iff the rays \overline{pξ_{1}} and \overline{pξ_{2}} bound a flat sector.

By the Cartan-Hadamard theorem [1, 3], the universal cover of a connected, complete, length space with nonpositive curvature is a Hadamard space with the natural metric. Let Z be a complete, connected space with nonpositive curvature, and let π : Z̃ → Z be the universal cover. If Y ⊂ Z is a closed, connected, locally convex subset, then the induced length metric on Y has nonpositive curvature, π^{−1}(Y) ⊂ Z̃ is a disjoint union of closed convex components isometric to Ỹ, and the induced map π_{1}(Y) → π_{1}(Z) is a monomorphism.

1.3 Leeb complexes

The following piecewise Euclidean 2-complexes were suggested to us by Bernhard Leeb, after a discussion of the graph manifold geometry in [6].

Let T_{0}, T_{1}, T_{2} be flat two-dimensional tori. For i = 1, 2, we assume that there are (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 once at an angle α ∈ (0, \frac{π}{2}]. The resulting 2-complex X̄ is nonpositively curved as a length space because gluing of nonpositively curved spaces along locally convex subsets produces a nonpositively curved space [3]. We refer to X̄ as a Leeb complex. For i = 1, 2 let Ȳ_{i} := T_{0} ∪ T_{i} ⊂ X̄. Notice that Ȳ_{i} and T_{0} are closed, locally convex subsets of X̄. Therefore the inclusions Ȳ_{i} ⊂ X̄ and T_{0} ⊂ X̄ induce monomorphisms of fundamental groups.

1.4 The structure of the universal cover

Let π : X → X̄ be the universal covering of X̄. X is a Hadamard space by the Cartan-Hadamard theorem. A block is a connected component of π^{−1}(Ȳ_{i}) ⊂ X, and a wall is a connected component of π^{−1}(T_{0}) ⊂ X. Let B and W denote the collection of blocks and walls in X. Each block (resp. wall) is a closed, connected, locally convex subset of X which is intrinsically isometric to the universal cover of Ȳ_{i} (resp. T_{0}) since the inclusion Ȳ_{i} → X̄ (resp. T_{0} → X̄) induces a monomorphism of fundamental groups. Hence each block (resp. wall) is a convex subset of X. If W ∈ W, B ∈ B, then either W ∩B = ∅ or W ∩B = W since W ∩B is open and closed in W; W is contained in precisely two blocks, one covering Ȳ_{1} and the other covering Ȳ_{2}. If B_{1}, B_{2} ∈ B are distinct blocks and B_{1} ∩ B_{2} ≠ ∅, then (after relabelling if necessary) B_{i} covers Ȳ_{i} and so B_{1} ∩ B_{2} consists of a (convex) union of walls; therefore B_{1} ∩ B_{2} = W for some W ∈ W. When B_{1} ∩ B_{2} ≠ ∅ we will say that the blocks B_{1} and B_{2} are adjacent.

Ȳ_{i} is a “flat” S^{1} bundle over a bouquet of two circles, so the universal cover Y_{i} of Ȳ_{i} (and hence each block) is isometric to the metric product of a simplicial tree with R. A singular geodesic of a block B is the inverse image of a vertex under the projection of B to its tree factor. Note that singular geodesics of adjacent blocks which lie in the common wall intersect at angle α.

B and W are clearly locally finite collections. The nerve of B (the simplicial complex recording (multiple) intersections of blocks) is a simplicial tree. To see this note that if ϵ > 0 is sufficiently small and B_{ϵ} is the collection of (open) ϵtubular neighborhoods of blocks, then Nerve(B_{ϵ}) is isomorphic to Nerve(B). Using a partition of unity subordinate to this cover of |Nerve(B_{ϵ})| one gets a continuous map ϕ : X → |Nerve(B_{ϵ})|. Any map γ : S^{1} → |Nerve(B)| can be “lifted” to X up to homotopy: there is a γ̂ : S^{1} → X so that p ∘ γ̂ is homotopic to γ. Since π_{1}(X) is trivial, this implies that π_{1}(|Nerve(B)|) is trivial. In particular, every wall separates X.

Our plan is to show that the subspace ∪_{B∈B} ∂_{∞}B ⊂ ∂_{∞}X can be characterized purely topologically^{3}, and that its topology is different depending on whether α = \frac{π}{2} or not. It will then follow that a Leeb complex with α < \frac{π}{2} and a Leeb complex with α = \frac{π}{2} have universal covers with nonhomeomorphic ideal boundaries.

1.5 Itineraries

For each p ∈ X \ ∪_{W ∈W} W, ξ ∈ ∂_{∞}X, we get a sequence of blocks B_{i} called the p-itinerary (simply the itinerary if the basepoint p is understood) of ξ, as follows.

3At first glance one might think that ∪_{B∈B} ∂_{∞}B is a path component of ∂_{∞}X, but this turns out not to be the case. It is a “safe” path component, see 1.7.

Let B_{i} be the i^{th} block that the ray \overline{pξ} enters; the ray enters a block B if it reaches a point in B \ ∪_{W ∈W}W. We will denote the p-itinerary of \overline{pξ} by Itin(\overline{pξ}) or Itin(ξ).

Lemma 2 The itinerary defines a geodesic segment or geodesic ray in the simplical tree Nerve(B).

Proof. Blocks are convex, so a geodesic cannot revisit any block which it left. The topological frontier of any B ∈ B is the union of the walls contained in B, so a geodesic segment which leaves B must arrive at a wall W ⊂ B, and then enter the block B′ ∈ B adjacent to B along W. The collection B is locally finite, so the lemma follows □

Note that ξ ∈ ∂_{∞}X has a finite itinerary iff ξ ∈ ∂_{∞}B for some B ∈ B.

1.6 Local components of ∂_{∞}X

Since each block B is isometric to the product of simplicial tree with R, ∂_{∞}B is homeomorphic to the suspension of a Cantor set. A pole of B is one of the two suspension points in ∂_{∞}B.

Lemma 3 If B_{1}, B_{2} ∈ B, then one of the following holds:

1. ∂_{∞}B_{1} ∩ ∂_{∞}B_{2} = ∅.

2. B_{1} ∩ B_{2} = W ∈ W and ∂_{∞}B_{1} ∩ ∂_{∞}B_{2} = ∂_{∞}W.

3. There is a B ∈ B such that B ∩ B_{i} = W_{i} ∈ W and ∂_{∞}B_{1} ∩ ∂_{∞}B_{2} is the set of poles of B.

Proof. Suppose B_{1}, B_{2} ∈ B are distinct blocks, ξ ∈ ∂_{∞}B_{1} ∩ ∂_{∞}B_{2}, and W ∈ W is a wall separating B_{1} from B_{2}. Choose basepoints b_{i} ∈ B_{i}, w ∈ W. If x_{k} ∈ \overline{b_{1}ξ} is a sequence tending to infinity, and y_{k} ∈ \overline{b_{2}ξ} is a sequence with d(y_{k}, x_{k}) < C, then we can find a z_{k} ∈ \overline{x_{k}y_{k}} ∩ W since W separates B_{1} from B_{2}. Therefore \overline{wz_{k}} ⊂ W converges, and the limit ray \overline{wξ} lies in W. Hence ξ ∈ ∂_{∞}W.

Note that if W_{1}, W_{2} ⊂ B ∈ B, then ∂_{∞}W_{1} ∩ ∂_{∞}W_{2} is just the set of poles of B; and ξ ∈ ∂_{∞}X cannot be a pole of two adjacent blocks simultaneously.

The lemma follows, since ∂_{∞}B_{1} ∩ ∂_{∞}B_{2} ≠ ∅ now implies that the combinatorial distance between B_{1} and B_{2} in Nerve(B) is ≤ 2. □

Lemma 4 Suppose ξ lies on the ideal boundary of a block B ∈ B, and assume ξ is not a pole of any block other than B. Then the path component of ξ in a suitable neighborhood Ω of ξ is contained in ∂_{∞}B.

Proof. Case I: ξ ∈ ∂_{∞}B is a pole of B. Choose p ∈ B \ ∪_{W ∈W}W. Recall (see section 1.3) that α is the angle between singular geodesics of adjacent blocks lying in the common wall, so α is the minimum Tits angle between ξ and any pole of a block adjacent to B. Let Ω := \{ξ′ ∈ ∂_{∞}X | ∠_{p}(ξ′, ξ) < \frac{α}{2}\}, where ∠_{p}(ξ, ξ′) is the angle between the initial velocities of the two rays \overline{pξ}, \overline{pξ′}. We define an exit from B to be a singular geodesic E ⊂ B of a block adjacent to B. A ray \overline{pξ′} exits from B via E if \overline{pξ′} ∩ B is a geodesic segment ending at E, and the ray \overline{pξ′} continues into the block containing E. For each exit E from B, let Ω_{E} be the set of ξ′ ∈ Ω such that \overline{pξ′} exits B via E.

Sublemma 5 Ω_{E} is an open and closed subset of Ω.

Proof. Openness. If ξ′ ∈ Ω_{E}, then \overline{pξ′} ∩ B is a segment ending at some e ∈ E, and \overline{pξ′} enters the block B′ adjacent to B which contains E. But then any sufficiently nearby ray \overline{pξ′′} also leaves B at a point close to e; clearly this point must lie on E. Therefore Ω_{E} is open in ∂_{∞}X.

Closedness. Let E′ ⊂ E be the set of “exit points” for elements of Ω_{E}: the endpoints of segments \overline{pξ′} ∩ B, where ξ′ ∈ Ω_{E}. E′ is bounded, for otherwise we could find a sequence e_{k} ∈ E′ with lim_{k→∞} d(e_{k}, p) = ∞, and get a limit ray \overline{pe_{∞}} ⊂ B with e_{∞} ∈ ∂_{∞}E ⊂ ∂_{∞}B ∩ ∂_{∞}B′, and ∠_{p}(ξ, e_{∞}) ≤ \frac{α}{2}; this is absurd since e_{∞} is a pole of B′ and so ∠_{p}(e_{∞}, ξ) = ∠_{T its}(e_{∞}, ξ) ≥ α. Now suppose ξ′_{k} ∈ Ω_{E} and lim_{k→∞} ξ′_{k} = ξ′_{∞} ∈ Ω. We have, after passing to a subsequence if necessary, that \overline{pξ′_{k}} ∩B = \overline{pe_{k}} where e_{k} ∈ E and lim_{k→∞} e_{k} = e_{∞} ∈ E. Then \overline{pξ_{∞}} ∩ B contains \overline{pe_{∞}} and clearly we have ξ′_{∞} ∈ Ω_{E}.

□

It follows that the connected (or path) component of ξ in Ω is contained in ∂_{∞}B, since any subset C ⊆ Ω containing ξ and intersecting Ω_{E} admits a separation C = (C ∩ Ω_{E}) ∪ (C \ Ω_{E}) into open subsets of C, and any ξ′ ∈ Ω \ ∂_{∞}B lies in Ω_{E} for some E. Case II: ξ ∈ ∂_{∞}W where W is the wall separating two adjacent blocks B_{1}, B_{2}, and ξ is not a pole. Pick p ∈ W not lying on a singular geodesic. Let ψ be the minimum Tits distance between ξ and a pole of B_{i}, i = 1, 2, and set

Ω := \{ξ′ ∈ ∂_{∞}X | ∠_{p}(ξ′, ξ) < \frac{ψ}{2} \}.

Let E be a singular geodesic of B_{1} or B_{2} which is contained in W. We say that the ray \overline{pξ′} exits W via E if \overline{pξ′} ∩ W ends at a point in E, and \overline{pξ′} then immediately enters the block corresponding to E. Let Ω_{E} be the set of ξ′ ∈ Ω so that \overline{pξ′} exits W via E. One checks as in case I that Ω_{E} is closed and open in Ω, so we conclude that the connected component of ξ in Ω is contained in ∂_{∞}W. Case III: ξ ∈ ∂_{∞}B does not lies in the boundary of any block other than B. Let ϕ be the minimum Tits angle between ξ and a pole of B, and set

Ω := \{ξ′ ∈ ∂_{∞}X | ∠_{p}(ξ′, ξ) < \frac{ϕ}{2}\}.

Pick p ∈ B \ ∪_{W ∈W}W. Since ξ is not a pole of B, the ray \overline{pξ} determines a half-plane H ⊂ B, the intersection of the flat planes in B containing it. Let B′ be the collection of blocks adjacent to B. If B′ ∈ B′ then B′ ∩ H (= W ∩ H where W = B ∩ B′ is the wall between B and B′) is either empty, a singular geodesic of B, or a flat strip with finite width bounded by singular geodesics, for otherwise we would have ξ ∈ ∂_{∞}B′. Removing the singular geodesics and ∪_{B′∈B′}B′ from H, we get a subset H^{0} whose connected components are a countably infinite collection of open strips. If S ⊂ H^{0} is such a strip, we let Ω_{S} be the set of ξ′ ∈ Ω so that \overline{pξ′} ∩ S ≠ ∅. As in cases I and II, Ω_{S} is closed and open in Ω. This forces the connected component of ξ in Ω to be contained in ∂_{∞}H ⊂ ∂_{∞}B, as desired. □

1.7 Vertices and safe paths

We say that ξ ∈ ∂_{∞}X is a vertex if there is a neighborhood U of ξ such that the path component of ξ in U is homeomorphic to the cone over a Cantor set, with ξ corresponding to the vertex of the cone. By Lemma 4 a pole of any block is a vertex (a priori there may be other vertices in ∂_{∞}X).

A path c : [0, 1] → ∂_{∞}X is safe if c(t) is a vertex for only finitely many t ∈ [0, 1].

Lemma 6 ∪_{B∈B} ∂_{∞}B is a safe path component of ∂_{∞}X.

Proof. First note that if c : [0, 1] → ∂_{∞}X is a path, c(t) is not a vertex when t ∈ (0, 1), B ∈ B, and c(0) ∈ ∂_{∞}B is not a pole of any block other than B, then c([0, 1]) ⊂ ∂_{∞}B. This follows from Lemma 4, the fact that ∂_{∞}B is closed in ∂_{∞}X, and a continuity argument.

Now if B_{0} ∈ B, c : [0, 1] → ∂_{∞}X is a safe path starting in ∂_{∞}B_{0}, and 0 = t_{0} < t_{2} … < t_{k} = 1 are chosen so that c(t) is a vertex only if t = t_{i} for some i, then one proves by induction on i that the intervals [t_{i−1}, t_{i}] are mapped into ∪_{B∈B} ∂_{∞}B. □

Lemma 7 Pick B_{0} ∈ B and p ∈ B_{0} \ ∪_{W ∈W}W. Let c : [0, 1] → ∂_{∞}X be a path, and suppose c(0) has an infinite p-itinerary. Then either c(t) has the same p-itinerary as c(0) for all t ∈ I, or there is a t̄∈ I so that c(t̄) has finite itinerary.

Proof. Suppose ξ_{k} ∈ ∂_{∞}X is a sequence with lim_{k→∞} ξ_{k} = ξ ∈ ∂_{∞}X, and a certain block B is in the itinerary of \overline{pξ_{k}} for every k. Then either

1. Itin(ξ) contains B

or

2. Itin(ξ) is finite and only contains blocks lying between B_{0} and B. To see this, suppose B′ is in Itin(ξ) and x ∈ \overline{pξ} ∩Int(B′). Then x = lim_{j→∞} x_{j} where x_{j} ∈ \overline{pξ_{j}} ∩ Int(B′) for sufficiently large j, so B′ is in Itin(ξ_{j}) for sufficiently large j. This means that B′ lies between B_{0} and B, for otherwise B would have to lie between B_{0} and B′, forcing B ∈ Itin(ξ).

The lemma now follows, since if B is in Itin(c(0)) but not in Itin(c(t)) for all t ∈ [0, 1], then settting t_{0} := inf\{t | B ∉ Itin(c(t))\} we get a ray \overline{pc(t_{0})} with finite itinerary by the reasoning of the preceding paragraph. □

Corollary 8 There is a unique safe path component of ∂_{∞}X which is dense, namely ∪_{B∈B} ∂_{∞}B.

Proof. By Lemma 6 we know that ∪_{B∈B} ∂_{∞}B forms a safe path component. ∪_{B∈B} ∂_{∞}B is dense in ∂_{∞}X since any initial segment \overline{px} of a ray \overline{pξ} may be continued as a ray \overline{pξ′} = \overline{px}∪\overline{xξ′} where the continuation \overline{xξ′} lies in a block (one of at most two) containing x.

By Lemma 7, if ξ ∈ ∂_{∞}X has an infinite p-itinerary, then any safe path starting at ξ consists of points with the same p-itinerary. Clearly the collection of points with a given p-itinerary isn’t dense in ∂_{∞}X. The corollary follows. □

1.8 Detecting block boundaries

Call an arc I ⊂ ∪_{B∈B} ∂_{∞}B an edge if its endpoints are both vertices, but no interior point of I is vertex of ∂_{∞}X. Edges are contained in the boundary of a single block B ∈ B (see the proof of Lemma 6). Clearly the endpoints of an edge I ⊂ ∪_{B∈B} ∂_{∞}B are either the poles of a single block, or I ⊂ ∂_{∞}W where W = B_{1} ∩ B_{2} and the endpoints of I are poles of B_{1} and B_{2}. So two points in ∪_{B∈B} ∂_{∞}B are the poles of a single block (resp. adjacent blocks) iff they are the endpoints of more than one edge (resp. a unique edge). A subset of ∪_{B∈B} ∂_{∞}B is the boundary of a block B ∈ B iff it is the union of all edges intersecting the poles of B.

1.9 Limiting behavior of poles

Pick B ∈ B, and consider the set P of poles of blocks adjacent to B. If η ∈ ∂_{∞}B is a pole of B, then we have ∠_{T its}(ξ, η) ∈ {α, π − α} for every ξ ∈ P. Let P_{α} := {ξ ∈ P | ∠_{T its}(ξ, η) = α}, and P_{π−α} := {ξ ∈ P | ∠_{T its}(ξ, η) = π − α}. Call each arc of ∂_{∞}B joining the poles of B a longitude.

Lemma 9 Each longitude of ∂_{∞}B intersects P̄_{α} (resp. P̄_{π−α}) in a single point ξ with ∠_{T its}(ξ, η) = α (resp ∠_{T its}(ξ, η) = π − α).

Proof. Pick p ∈ B, ξ ∈ ∂_{∞}B with ∠_{T its}(ξ, η) = α. Any initial segment \overline{px} of the ray \overline{pξ} may be extended to a segment \overline{py} = \overline{px} ∪ \overline{xy} so that \overline{py} ∩ W = \{y\} for some wall W ⊂ B. Then \overline{py} may be extended as a ray \overline{pξ′} = \overline{py} ∪ \overline{yξ′} where \overline{yξ′} ⊂ W and ξ′ ∈ P_{α}. Therefore ξ ∈ P̄_{α}. Since ∠_{T its}(·, η) is a continuous function on ∂_{∞}B, each longitude intersects P_{α} in a single point. Similar reasoning applies to P_{π−α}. □

From the lemma we see that any longitude l of ∂_{∞}B intersects P̄ in two points if α < \frac{π}{2} and one point if α = \frac{π}{2}.

1.10 Distinguishing Leeb complexes

Let X̄_{1} be a Leeb complex with α < \frac{π}{2}, and let X̄_{2} be a Leeb complex with α = \frac{π}{2}. Let X_{1} and X_{2} be their respective universal covers. A homeomorphism f : ∂_{∞}X_{1} → ∂_{∞}X_{2} would carry safe path components to safe path components, block boundaries to block boundaries (Corollary 8 and section 1.8), poles to poles, and longitudes to longitudes. But then section 1.9 gives a contradiction.

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