Conjugacy rigidity for nonpositively curved graph manifolds

Christopher B. Croke^{∗}

June 28, 2001

Abstract

We show that the metric of nonpositively curved graph manifolds is determined by its geodesic flow. More precisely we show that if the geodesic flows of two nonpositively curved graph manifolds are C^{0} conjugate then the spaces are isometric.

1 Introduction

Two complete Riemannian manifolds, N and N′, are said to have C^{0} conjugate geodesic flows if there is a continuous map F : S^{1}N → S^{1}N′ such that F ∘ g_{t} = g′_{t} ∘ F for all t where g_{t} (resp. g′_{t}) represents the geodesic flow on the unit sphere bundle S^{1}N (resp. S^{1}N′). The conjugacy is called C^{k} if F is C^{k}.

Graph manifolds of nonpositive curvature are compact three dimensional manifolds whose geometry had been extensively studied in [Sh] and [Le]. Though there have been a number of definitions of ”graph manifold” we will use a the rather general one used in [Le]: a graph manifold is a 3 dimensional Haken manifold each of whose components under the minimal decomposition (in the sense of [J-S, Jo]) is Seifert fibered. The decomposition is by cutting along embedded incompressible tori and Klein bottles. Not all such manifolds admit metrics of nonpositive curvature (see [Le]) but this paper is only concerned with the nonpositively curved Riemannian setting.

They are of interest for a number of reasons, but in particular they are the easiest nontrivial examples of nonpositively curved manifolds that have Ballmann rank 1 but whose fundamental group is not hyperbolic. Here we show:

Theorem 1.1. If N and N′ are compact nonpositively curved graph manifolds whose geodesic flows are C^{0} conjugate then they must be isometric.

The conjugacy rigidity question for compact Riemannian manifolds: “When does a C^{k} conjugacy between the geodesic flows of two compact Riemannian manifolds N and N′ imply that N and N′ are isometric?” is a much studied but still largely open question. This question is closely related to other rigidity problems (such as the ”boundary rigidity” problem and - in negative curvature - the ”marked length spectrum” rigidity problem).

∗Supported by NSF grant DMS 99-71749 and a CRDF grant.

There are many examples where isometry does not hold (Zoll surfaces or more generally the examples in section 6 of [C-K2] for example) however none of these occur in the nonpositive curvature setting. The only general conjugacy rigidity result known where there are no curvature assumptions made is that of [C-K2] where one of the metrics is assumed to have a parallel vector field (hence it is locally a Riemannian product where one factor is an interval). A natural setting for these questions is when at least one of the manifolds (say N) is assumed to have nonpositive curvature. The two dimensional case (when N has nonpositive curvature) is completely taken care of by [C-F-F], which is based on [O], and [Cr1]. In higher dimensions, if N is a locally symmetric space of negative curvature (see [B-C-G]) or a flat manifold (see [Cr2]) and if the conjugacy is C^{1} (or C^{0} if V ol(N) = V ol(N′)) then N′ is isometric to N.

In this paper we consider the case where both manifolds N and N′ have nonpositive curvature and n ≥ 3 (since the 2 dimensional case was taken care of by [C-F-F]). In this setting it is known that N′ must be isometric to N if N has Ballmann rank ≥ 2 (see [C-E-K]).

In the above I restricted my attention only to rigidity theorems for a pair of compact manifolds without boundary. There are a number of results on compact manifolds with boundary or on finite volume manifolds. Much work has also been done on the deformation question (i.e. can there be nontrivial differentiable one parameter families N_{t} with the ”same” geodesic flows) (see for example [G-K], [C-S], [D-S], and [S-U]).

In a previous paper, [C-K1], motivated by a question of Gromov in [Gr], the author along with Bruce Kleiner considered the space of geodesic rays on the universal cover of such nonpositively curved graph manifolds (as well as other related spaces). The asymptotic properties of the geodesic rays are encoded in the geodesic boundary (equivalence classes of rays) and the induced action of the fundamental group on that boundary. We were able to give necessary and sufficient conditions on the metrics to guarantee that two such (necessarily homeomorphic) spaces have universal covers with equivalent boundaries (i.e. there is a an equivariant, with respect to the action of the fundamental group, homeomorphism between the boundaries). In the process we saw that there were examples of non isometric graph manifolds of nonpositive curvature whose universal covers have equivalent boundaries. It thus became natural to ask if complete knowledge of the geodesic flow (i.e. knowledge up to C^{0} conjugacy) is enough to completely determine the metric. (Such a conjugacy, of course, induces an equivalence between boundaries in a natural way.) Theorem 1.1 gives a positive answer to that.

Graph manifolds of nonpositive curvature have a very restricted geometric form as studied in [Sh]. They are the union over totally geodesic boundaries of pieces (the “geometric Seifert components”) whose universal covers are isometric to a Riemannian product X ×ℝ where X is a surface of nonpositive curvature with geodesic boundary. The universal cover of the geometric Seifert components are minimal displacement sets (see section 2 for a definition) of isometries. In fact, the minimal displacement set of an isometry of any simply connected space of nonpositive curvature has the form X × ℝ where X is a convex subspace. They thus become amenable to some techniques developed in [C-K2]. In sections 2 and 3 of the paper we thus consider C^{0} geodesic conjugacies between arbitrary nonpositively curved manifolds of dimension ≥ 3. In section 3 we show that for corresponding minimal displacement sets the conjugacy induces affine maps between the corresponding spaces X (see Proposition 3.3).

In section 4 we restrict our attention to graph manifolds and prove Theorem 1.1. Using the results in section 3 it is straightforward to show that corresponding Seifert components are isometric. However, we note that it is easy to glue isometric Seifert components together so as to make non isometric spaces. The main point of section 4 is to show that the components must glued together in an isometric manner.

We are grateful to Bruce Kleiner for helpful comments on earlier versions of this paper.

2 Notation and preliminaries

A standard consequence of the flat strip theorem says that if ν is a geodesic in a simply connected nonpositively curved manifold M then the space M_{ν} = {x ∈ M|x lies on a geodesic that stays a bounded distance from ν} is a convex subset of M which is isometric to X_{ν} ×ℝ where X_{ν} is a simply connected manifold of nonpositive curvature with convex boundary. The unit speed geodesics that stay a bounded distance from ν are precisely the ”vertical” curves t → (x_{0}, t + t_{0}) ∈ X_{ν} × ℝ for fixed (x_{0}, t_{0}). We will let π_{ν} : M_{ν} → X_{ν} (resp. ℝ_{ν} : M_{ν} → ℝ) be the projection.

Let N be a Riemannian manifold of dimension ≥ 3 and of nonpositive curvature and π_{N} : M → N its universal cover. Since basepoint projection, b : S^{1}N → N, induces an isomorphism between π_{1}(S^{1}N) and π_{1}(N) ≡ Γ we see that π_{N∗} : S^{1}M → S^{1}N is also a universal cover. (For any tangent vector v to any space we will let b(v) represent its base point.) After appropriate choices of base points we get cocompact actions of Γ on M and S^{1}M with M/Γ = N and S^{1}M/Γ = S^{1}N. For every γ ∈ Γ we let M_{γ} ⊂ M be the set of points of minimal γ-displacement, i.e. the set of m such that d(m, γ(m)) is a minimum. This is exactly the space M_{ν} where ν is any axis of γ (i.e. γ(ν) = ν) and the axes of γ are precisely the vertical geodesics in Mγ. Thus M_{γ} is a convex subset of M which is isometric to X_{γ} × ℝ where γ acts by translation on the ℝ factor and X_{γ} is a simply connected manifold of nonpositive curvature with convex boundary. (We will use π_{γ} : M_{γ} → X_{γ} and ℝ_{γ} : M_{γ} → ℝ) to represent the projections.) The centralizer, Z(γ) ⊂ Γ, of γ in Γ, will act cocompactly on M_{γ} preserving the vertical directions.

Note that the geometric Seifert components of a nonpositively curved graph manifold are of the form M_{γ}/Γ_{γ} where Γ_{γ} is the subset of Γ that preserves M_{γ}. Thus Z(γ) ⊂ Γ_{γ}, but Γ_{γ} may also contain elements which reverse the vertical direction.

For every m ∈ M_{γ}, π_{N} takes the (vertical) geodesic segment \overline{mγ(m)} to a closed geodesic in the free homotopy class of γ, and all such closed geodesics come this way.

3 Geodesic conjugacies and the M_{ν}

In this section we consider compact nonpositively curved manifolds N and N′ of dimension ≥ 3. We discuss the consequences that a C^{0} conjugacy of the geodesic flows has on the spaces M_{ν} (and in particular on the M_{γ}).

The isomorphism F_{∗} : π_{1}(S^{1}N) → π_{1}(S^{1}N′), induces an isomorphism between π_{1}(N) and π_{1}(N′). We will let Γ represent the common abstract group. Thus (after choices of base points compatible with F for M M′, N, and N′) we can lift F to a uniformly continuous conjugacy F̃ : S^{1}M → S^{1}M′ and have cocompact actions of Γ on M and M′ whose differentials commute with F̃. We will let D(F) < ∞ be the maximum over m ∈ M of the diameters of the sets F̃(S_{m}^{1}M) ⊂ S^{1}M′ where S_{m}^{1}M ⊂ S^{1}M represents the unit vectors with basepoint m.

Lemma 3.1. For all v, w ∈ S^{1}M:

−2D(F) ≤ d_{M}(b(v), b(w)) − d_{N}(b(F̃(v)), b(F̃(w))) ≤ 2D(F)

Proof: For v, w ∈ SM, let τ : [0, l] → M be a minimizing geodesic from b(v) to b(w). Then both d_{N}(b(F̃(v)), b(F̃(τ′(0)))), and d_{N}(b(F̃(w)), b(F̃(τ′(l)))), are bounded by D(F) and d_{N}(b(F̃(τ′(0))), b(F̃(τ′(l)))) = d_{N}(b(F̃(τ′(0))), b(g^{l}(F̃(τ′(0))))) = l = d_{M}(b(v), b(w)). Hence the inequalities follow from the triangle inequality. □

For any geodesic ν of M, F̃ will take the unit tangent vector field of ν to the unit tangent field of a geodesic of M′ which by abuse of notation we will call F̃(ν).

For ν a geodesic in M and m ∈ M_{ν} we let V (m) be the vertical direction at m (i.e. points in the positive ℝ direction of M_{ν} = X_{ν} ×ℝ). An immediate consequence of the above lemma is that if a geodesic ν_{1} stays a bounded distance from ν then F̃(ν_{1}) stays a bounded distance from F̃(ν). Thus F̃(V (m)) = V′(m′) for some m′ ∈ M′_{F̃(ν)}. This

allows us to define a map G_{ν} : M_{ν} → M′_{F̃(ν)} with F̃(V (m)) = V′(G_{ν}(m)), and then, since we have G_{ν}(τ_{t}m) = τ′_{t}(G_{ν}(m)) (where τ_{t} and τ′_{t} represent vertical translation in M_{ν} = X_{γ} ×ℝ and M′_{ν} respectively), we get a well defined function GX_{ν} : X_{ν} → X′_{F̃(ν)}.

For γ ∈ Γ the above applies to M_{γ}. Since F must take closed geodesics in N in the free homotopy class of γ to corresponding closed geodesics (of the same length) in N′ we see that F̃ takes M_{γ} to M′_{γ} and gives rise to maps G_{γ} and GX_{γ} as above. This also tells us that the translation by γ on M_{γ} and M′_{γ} is by the same amount. We also note that since F̃ commutes with the differentials of Z(γ), and since Z(γ) preserves M_{γ} and vertical directions, we have that G_{γ} commutes with Z(γ) (which acts cocompactly).

A curve c in M_{ν} = X_{ν} × ℝ is said to have ℝ growth rate A if lim_{t→∞}(\frac{ℝ_{ν}(c(t))}{t}) = A.

Lemma 3.2. If a curve c in M_{ν} has growth rate A then G_{ν}(c) also has ℝ growth rate A.

Proof: Let B_{ν} be the Busemann function of ν (i.e. B_{ν}(x) = lim_{t→∞}(t − d(x, ν(t))). Now Lemma 3 (with both v and w vertical vectors) implies that |B_{ν}(x) − B_{F̃(ν)}(G_{ν}(x))| ≤ 2D(F). On the other hand B_{ν}(x) = ℝ_{ν}(x) + const and hence ℝ growth rates are the same as Busemann growth rates (also in M′_{F̃(ν)}) and the lemma follows.

□

Proposition 3.3. For any geodesic ν in M, GX_{ν} takes each unit speed geodesic segment in X_{ν} to a constant speed geodesic segment in X′_{F̃(ν)}. Further, if the segment is a subset of a ray in X_{ν} then the image is unit speed.

Proof:

Let σ(t) for 0 ≤ t ≤ L be a unit speed geodesic segment in X_{ν} and for each i let σ_{i}(t) = (σ(\frac{t}{\sqrt{1+i^{2}}} ), \frac{it}{\sqrt{1+i^{2}}} )) ⊂ M_{ν} for 0 ≤ t ≤ L\sqrt{1 + i^{2}} be a unit speed geodesic in M_{ν}. Now F (σ′_{i}(t)) will be the tangent vector field, τ′(t), to a unit speed geodesic τ_{i}(t) in M′. Let p_{i} ∈ M′_{F̃(ν)} (resp. q_{i} ∈ M′_{F̃(ν)}) be the unique closest point in M′_{F̃(ν)} to τ_{i}(0) (resp. τ_{i}(L\sqrt{1 + i^{2}})). The uniform continuity of F guarantees d(F (σ′_{i}(t)), F (V (σ_{i}(t)))) converges to 0 (d computed in the tangent bundle) and hence that lim_{i→∞} d(p_{i}, τ_{i}(0)) = 0 and lim_{i→∞} d(q_{i}, τ_{i}(L\sqrt{1 + i^{2}})) = 0. Hence if τ̄_{t}(t) for 0 ≤ t ≤ L\sqrt{1 + i^{2}} is the constant speed geodesic from p_{i} to q_{i} we see (by the nonpositivity of the curvature) that for every t, d(τ′_{i}(t), τ̄′_{i}(t)), and hence d(τ̄′_{i}(t), F (V (σ_{i}(t)))), converges to zero. In particular for every t we have d(π′_{F̃(ν)}(τ̄′_{i}(t)), GX_{ν}(σ(\frac{t}{\sqrt{1+i^{2}}} ))) converges to 0. But each π′_{F̃(ν)}(τ̄′_{i}(t)) is a constant speed geodesic in X′_{F̃(ν)} and hence so is GX_{ν}(σ(t)).

Now let r; [0, ∞) → X_{ν} be a unit speed ray and assume that GX_{ν}(r) is a ray with constant speed a. We need to show that a = 1. Using the construction above for segments, we first define (by letting L go to ∞) for each i the unit speed ray τ_{i}(t) whose distance to M′_{F̃(ν)} goes uniformly (since F̃ is uniform) to 0 as i goes to infinity and hence we can define (by a standard limiting argument) a ray τ̄_{i}(t) ⊂ M′_{F̃(ν)} such that as i goes to ∞ both d(τ̄′_{i}(t), τ′_{i}(t)) and d(π′_{F̃(ν)}(τ̄′_{i}(t)), GX_{ν}(r(\frac{t}{\sqrt{1+i^{2}}} ))) converge uniformly to 0. Note that d(τ̄_{i}(t), G_{ν}(σ_{i}(t)) is uniformly bounded since both curves are at a uniformly bounded distance (depending on D) from τ_{i}(t), and hence, by Lemma 3.2, τ̄_{i} has the same ℝ growth rate as σ_{i} (i.e.\frac{i}{\sqrt{1+i^{2}}} ). But since GX_{ν}(r) has speed a (and hence GX_{ν}(r(\frac{t}{\sqrt{1+i^{2}}} )) has speed \frac{a}{\sqrt{1+i^{2}}} ), and τ̄_{i} has speed approaching 1 (the speed of τ_{i}) we see that τ̄_{i} can have the same ℝ growth as σ_{i} for all i if and only if a = 1. □

Maps GX_{ν} that satisfy the conclusion of Lemma 3.3 are called affine maps. We now recall some of the standard properties of such maps (e.g. see chapter VI of [K-N]). For V ∈ S^{1}X_{ν} let s(V) represent the speed of the GX_{ν} image of the geodesic determined by V. Since s is continuous it is locally bounded above and hence GX_{ν} is locally Lipschitz (in fact globally in the cocompact case). Thus GX_{ν} is differentiable almost everywhere and at every x ∈ X_{ν} where GX_{ν} is differentiable we see GX_{ν} = exp_{GX_{ν}(x)} ∘ D(GX_{ν})_{x} ∘ exp_{x}^{−1} (this is always true locally, but in our case exp_{x}^{−1} is globally defined). In particular GX_{ν} is C^{∞} and GX_{ν} = exp_{GX_{ν}(x)} ∘ D(GX_{ν})_{x} ∘ exp_{x}^{−1} holds for all x. This implies that GX_{ν} preserves the respective connections and hence that s is invariant under holonomy. Since X_{ν} is convex (with boundary) and simply connected, invariant subspaces of the holonomy action induce deRham splittings as usual. Thus we get:

Corollary 3.4. For any geodesic ν in M, GX_{ν} is C^{∞}, takes deRham factors to deRham factors, acts homothetically on each non Euclidean deRham factor (and hence is an isometry on each noncompact one), and acts as an affine transformation on the flat factor. In particular if X_{ν} is irreducible, not flat, and noncompact (e.g. in the case of M_{γ} when Z(γ) is not cyclic) then GX_{ν} is an isometry.

Remark 3.5. It is certainly possible that in the above GX_{ν} is always an isometry. In fact, it is highly likely if the conjugacy rigidity conjecture for nonpositively curved manifolds is true since in this case X_{ν} and X′_{F̃(ν)} would be isometric.

Corollary 3.6. When GX_{ν} is an isometry then G_{ν} : M_{ν} → M′_{ν} is also an isometry.

Proof. We know that G_{ν} : X_{ν}×ℝ → X′_{ν}×ℝ is of the form G_{ν}(x, t) = (GX_{ν}(x), t+f(x)) for some real valued function f : X_{ν} → ℝ. We need only see that f is constant. Along a geodesic segment σ (using notations from the proof of Proposition 3.3) we see that G_{ν}(σ_{i}(t)) gets uniformly close to the geodesic τ̄_{i}(t) as i gets large. This says that f must be linear along σ, but in fact it must be constant for as i gets large τ̄_{i}(t) becomes unit speed.

□

Remark 3.7. Consider the case that N is 3 dimensional, and for some γ ∈ Γ, X_{γ} is noncompact. Combining lemma 3.3 and the corollaries we see that either X_{γ} is a flat strip or G_{γ} is an isometry that commutes with Z(γ) and M_{γ}/Z(γ) is isometric to M′_{γ}/Z(γ).

Let O ⊂ X_{γ} be a flat strip neighborhood of some boundary geodesic τ, and Ō ⊂ M_{γ} be π_{γ}^{−1}(O), which has the metric of a flat plane cross an interval. Let α ∈ Z(γ), α ≠ γ be such that π_{γ}^{−1}(τ) ⊂ M_{γ} ∩ M_{α} then the geometry of Ō forces Ō ⊂ M_{γ} ∩ M_{α}.

4 Graph Manifolds

We now consider the case where N = M/Γ is a compact 3-dimensional graph manifold with a nonpositively curved Riemannian metric with fundamental group Γ and universal cover M.

By [Sh] and [Le], N has the following structure. There is a finite connected graph G with vertex set V and oriented edge set E (i.e. each edge shows up twice in E). For each v ∈ V There is a compact nonpositively curved 3-manifold N_{v} with nonempty totally geodesic boundary (the geometric Seifert components of N), and Seifert fibrations p_{v} : N_{v} → X_{v} where the metric on N_{v} has local product structure compatible with the fibration p_{v}, and the X_{v} are nonpositively curved orbifolds; N is the union of the N_{v}. Each N_{v} is M_{γ_{v}}/Γ_{γ_{v}} for some γ_{v} ∈ Γ. There is one boundary component T_{e} of N_{v} for each e ∈ E such that v is the initial vertex of e. (Note that a nonoriented edge from v to v will correspond to two boundary components of N_{v}.) The N_{v} are glued together as is described by G. That is for each edge e ∈ E, say from v_{1} to v_{2}, we glue N_{v_{1}} to N_{v_{2}} along T_{e} and T_{−e} respectively. (Note in the case where the metric on N_{v_{1}} near T_{e} is a product of an interval with T_{e} then remark 3.7 says that the picture looks the same on N_{v_{2}} near T_{−e} and the gluing identifies the whole product neighborhoods.) We will distinguish N_{v} of “oriented type” where Γ_{γ_{v}} = Z(γ_{v}) from the alternative “non-oriented type”.

We note that these gluings are only unique up to translations in the vertical directions of N_{v} of oriented type. In fact different choices of gluings lead to nonisometric manifolds (we explain more below).

It is easier to explain what is going on in the universal covering space. The universal covering space is built by gluing copies of the M_{γ_{v}}. The gluing is described by a tree Ḡ (the Basse-Serre tree of the fundamental group - see section 2.5 of [C-K1]) with vertex set V̄ and edge set Ē. Γ acts on Ḡ with π_{G} : Ḡ → Ḡ/Γ = G. We can define the vertex and edge spaces Mv̄, R2 ; ē in the universal cover, where Mv̄ is isometric to M_{γ_{v}} if π_{G}(v̄) = v. For every s ∈ ℝ there is an isometry τ(v̄, s) : Mv̄ → Mv̄ which translates by s in the vertical direction. (The notation τ(v̄, s) is somewhat unusual - rather than τ s ; v̄ for example - but it makes things easier to see later.) It is clear that τ(v̄, s) commutes with elements of Z(γ_{v}), however in the non-oriented case (unless s=0) it will not commute with those isometries that reverse the vertical orientation.

Now let N and N′ be nonpositively curved graph manifolds built from the same graph G and isometric pieces N_{v} and N′_{v} and let I_{v} be a chosen isometry respecting the vertical directions. We can lift the I_{v}’s equivariantly to isometries Iv̄ : Mv̄ → M ′ ; v̄. For each e ∈ Ē from v̄_{1} to v̄_{2} then Iv̄_{1} and Iv̄_{2} define two isometries from R2 ; ē to R^{2}′ ; ē which differ by a Euclidean translation. We can thus (after choosing upwards vertical directions in the N_{v}’s) uniquely define real numbers sē and rē such that on R2 ; ē we have

τ(v̄_{1}, sē) ∘ Iv̄_{1} = τ(v̄_{2}, rē) ∘ Iv̄_{2}.

We note that rē = s_{−ē} and that in the oriented case the equivariance of the lifting says that if π_{G}(ē_{1}) = π_{G}(ē_{2}) then sē_{1} = sē_{2} so we can define s_{e} for e ∈ E. In the non-oriented case one cannot choose the upwards vertical direction equivariantly so s_{e} is only defined up to sign and if ē_{1} and ē_{1} are edges from v and are related by an isometry that reverses the vertical direction of N_{v} then sē_{1} = −sē_{2} (independent of the choice of upwards).

We will call the relative gluings consistent if for every v ∈ G there is a s_{v} ∈ ℝ such that for every e ∈ E which has v as initial point we have s_{v} = s_{e}, and further if for N_{v} of non-oriented type s_{v} = 0 . If the gluings are consistent then we define an equivariant isometry I : M → M′ by I(m) = τ(v̄, sv̄) ∘ Iv̄(m) for m ∈ Mv̄. Thus we see:

Lemma 4.1. If N and N′ are nonpositively curved graph manifolds built from the same graph G and isometric pieces N_{v} and with consistent relative gluings then N and N′ are isometric.

Remark 4.2. If the relative gluings are not consistent then it is not hard to see there will be no isometry which takes N_{v} to N′_{v}. Thus we can make many nonisometric spaces with the same G and isometric pieces.

Two such nonisometric graph manifolds with the same G, and isometric N_{v} all of orientable type, but different gluings will have the same “geometric data” in the sense of [C-K1] and hence give a new class of examples of nonisometric graph manifolds whose geometric boundaries are equivariantly homeomorphic.

In order to make sure that in the non-oriented case our isometries G_{γ_{v}} : M_{γ_{v}} → M′_{γ_{v}} descend to N_{v} we need to make sure that G_{γ_{v}} commutes with the full group Γ_{γ_{v}} not just the subgroup Z(γ_{v}). In fact, this will not a-priori be the case. However we will see in the next lemma that for each such v there is a number a_{v} such that this holds if we replace F by the conjugacy g′_{a_{v}} ∘ F. This will allow us to define a Γ invariant (though a-priori not continuous) isometry from M to M′.

Lemma 4.3. For each v of non-oriented type there is an a_{v} such that if we define G_{γ_{v}} via the conjugacy g′_{a_{v}} ∘ F then G_{γ_{v}} commutes with the full group Γ_{γ_{v}}.

Proof. If instead of using the “upward” vertical direction of M_{γ_{v}} we used the “downward” vertical direction we would arrive at isometries Ḡ_{v} : M_{γ_{v}} → M′_{γ_{v}} and ḠX_{v} : X_{v} → X′_{v}. When X_{v} is not a flat strip the geometric structure of X_{v} guarantees that the identity is the only self isometry of X_{v} that takes each boundary component to itself. Hence we see that ḠX_{v} = GX_{v} and thus that Ḡ_{v} differs from G_{v} by at most a fixed vertical translation. Thus there is a (unique) a_{v} such that for vertical vectors V we have g′_{a_{v}} ∘ F (−V) = −g′_{a_{v}} ∘ F (v). This fact along with the fact that the conjugacy g′_{a_{v}} ∘F is Γ equivariant (here Γ acts on S^{1}M and S^{1}M′ via its differentials) easily yields the result since if ν ∈ Γ_{γ_{v}} reverses the orientation of the vertical then G ∘ ν = ν ∘ Ḡ = ν ∘ G.

We are assuming that the Seifert components are not products with an interval and hence the only case left to consider is when X_{v} = [0, L]×ℝ is a flat strip and there is an element ν ∈ Γ_{γ_{v}} such that ν reverses the vertical direction in M_{v} and switches boundary components. Thus the action of ν on M′_{v} = [0, L] × ℝ × ℝ (using the fact that there are no fixed points) is of the form ν(x, s, t) = (L − x, a + s, b − t) for some numbers a ≠ 0 and b. A similar argument to the above yields an a_{v} such that using g′_{a_{v}} ∘ F instead of F we have ḠX = T ∘ GX where T : [0, L] × ℝ → [0, L] × ℝ is of the form T (x, s) = (x, s_{0} ± s) and Ḡ = (T × id) ∘ G. We first note that the equivariance of the conjugacy F implies G ∘ ν = ν ∘ Ḡ = ν ∘ (T × id) ∘ G. Since both Ḡ and G are Z(γ_{v}) equivariant we see that T × id commutes with the Z(γ_{v}) action on M′_{v} which forces T to be of the form T (x, s) = (x, s_{0} +s). Now since ν^{2} preserves the orientation of the fibers we see that ν^{2} commutes with G and hence ν^{2} = ν ∘(T ×id)∘ν ∘(T ×id).

Thus (x, 2a + s, t) = (x, 2a + 2s_{0} + s, t) and so s_{0} = 0 and T = id. Thus Ḡ = G and the result follows as in the other case.

□

Proof of theorem 1.1 To prove Theorem 1.1 we use Remark 3.7 and Lemma 4.3 to define our isometries Iv̄ : Mv̄ → M ′ ; v̄ and assume (using Lemma 4.1) that N and N′ do not have consistent gluings. We will then find a closed geodesic c ∈ N and a curve ξ ∈ N′ representing the same (as identified via the conjugacy F) free homotopy class such that L(ξ) < L(c). This is the desired contradiction since any closed geodesic minimizes the lengths of curves freely homotopic to it and since ξ is freely homotopic to F (c) hence L(c) = L(F (c)) ≤ L(ξ). (We will do this by choosing c so that it intersects appropriate boundary tori nearly tangentially and such that the discontinuous image of c when we reglue the Seifert components to get N′ can be shortened to get a continuous curve in the appropriate homotopy class. We actually work in the universal covers.)

We first consider the case where v is of oriented type and we assume that there are edges e_{1}, e_{2} ∈ E and a vertex v such that v is the final point of e_{1} and the initial point of e_{2} and that s_{−e_{1}} ≠ s_{e_{2}}. We will let v_{1} be the initial point of e_{1} and v_{2} the final point of e_{2}.

We first consider the case where N_{v} is of oriented type. The first step is to find an appropriate closed geodesic c in N_{v_{1}} ∪ N_{v} ∪ N_{v_{2}} such that c intersects each of T_{e_{1}} and T_{e_{2}} in exactly two points and intersects no other T_{e}. To choose such a c we first choose closed geodesics a_{i} ⊂ N_{v_{i}} and b ⊂ N_{v}. We also choose minimizing geodesics τ_{i} from the base point of a_{i} to the base point of b. They are chosen so that b is in the homotopy class of γ_{v} and the classes corresponding to a_{i} (or more precisely −τ_{i} ∪ a_{i} ∪ τ_{i}) do not commute with γ_{v}. For each n ∈ ℤ we consider the curves

c_{n} = a_{1} ∪ τ_{1} ∪ b^{n} ∪ −τ_{2} ∪ a_{2} ∪ τ_{2} ∪ b^{−n} ∪ −τ_{1}.

We will choose c to be the closed geodesic in the free homotopy class of some c_{n} (for some n large in absolute value). Since c can be gotten from c_{n} by curve shortening, the convexity of the Seifert components can be used to make sure that each curve in the shortening process, and hence the final curve c, satisfies the required conditions.

Thus, we can choose a unit speed parameter c : [0, L] → N so that there are t_{i} ∈ R, 0 = t_{0} < t_{1} < t_{2} < t_{3} < t_{4} < t_{5} = L with c([0, t_{1}]) ⊂ N_{v_{1}}, c(t_{1}) ∈ T_{e_{1}}, c([t_{1}, t_{2}]) ⊂ N_{v}, c(t_{2}) ∈ T_{e_{2}}, c([t_{2}, t_{3}]) ⊂ N_{v_{2}}, c(t_{3}) ∈ T_{e_{2}}, c([t_{3}, t_{4}]) ⊂ N_{v}, c(t_{4}) ∈ T_{e_{1}}, c([t_{4}, L]) ⊂ N_{v_{1}}.

For such a c we will let c̄[0, l] → M be a continuous lift of c. There are thus v̄_{1}, v̄_{2}, v̄_{3}, v̄_{4}, v̄_{5} ∈ V̄ and ē_{1}, ē_{2}, ē_{3}, ē_{4} ∈ Ē such that for

i = 1, 2, 3, 4, 5, c̄_{i} ≡ c̄([t_{i−1}, t_{i}]) ⊂ Mv̄_{i},

i = 1, 2, 3, 4, c(t_{i}) ∈ ℝ2 ; ē_{i},

π_{G}(v̄_{1}) = π_{G}(v̄_{5}) = v_{1},

π_{G}(v̄_{2}) = π_{G}(v̄_{4}) = v,

π_{G}(v̄_{3}) = v_{2},

π_{G}(ē_{1}) = −π_{G}(ē_{4}) = e_{1},

and

π_{G}(ē_{2}) = −π_{G}(ē_{3}) = e_{2}.

We will now determine which n to use. Let ℝv̄_{i} : M_{v_{i}} → ℝ represent the projection onto the vertical. Choose |n| large enough that |ℝv̄_{2}(c̄(t_{2})) − ℝv̄_{2}(c̄(t_{1}))| > |sē_{2} − s_{−ē_{1}}| (resp. |ℝv̄_{4}(c̄(t_{4})) − ℝv̄_{4}(c̄(t_{3}))| > |sē_{2} − s_{−ē_{1}}| = |sē_{4} − s_{−ē_{3}}|). Choose the sign of n (hence the signs of ℝv̄_{2}(c̄(t_{2})) − ℝv̄_{2}(c̄(t_{1})) and ℝv̄_{4}(c̄(t_{4})) − ℝv̄_{4}(c̄(t_{3})) for large enough n) so that

|ℝv̄_{2}(c̄(t_{2})) − ℝv̄_{2}(c̄(t_{1})) + sē_{2} − s_{−ē_{1}}| < |ℝv̄_{2}(c̄(t_{2})) − ℝv̄_{2}(c̄(t_{1}))|

and

|ℝv̄_{4}(c̄(t_{4})) − ℝv̄_{4}(c̄(t_{3})) + sē_{4} − s_{−ē_{3}}| < |ℝv̄_{4}(c̄(t_{4})) − ℝv̄_{4}(c̄(t_{3}))|.

Having now chosen c we now proceed with the rest of the argument. We let γ ∈ Γ be such that γ(c̄(0)) = c̄(L), and hence in particular

γτ(v̄_{1}, sē_{1})(c̄(0)) = τ(v̄_{5}, sē_{1})(c̄(L)) = τ(v̄_{5}, s_{−ē_{4}})(c̄(L))

.

Consider the following continuous curve, ξ in M′ between τ(v̄_{1}, sē_{1})(c̄(0)) and τ(v̄_{5}, s_{−ē_{4}})(c̄(L)):

For t_{0} ≤ t ≤ t_{1}, ξ(t) = τ(v̄_{1}, sē_{1})(c̄_{1}(t)).

For t_{1} ≤ t ≤ t_{2}, ξ(t) = τ(v̄_{2}, \frac{t_{2}−t}{t_{2}−t_{1}} s_{−ē_{1}} + \frac{t−t_{1}}{t_{2}−t_{1}} sē_{2})(c̄_{2}(t)).

For t_{2} ≤ t ≤ t_{3}, ξ(t) = τ(v̄_{3}, sē_{3})(c̄_{3}(t)) (note sē_{3} = s_{−ē_{2}}).

For t_{3} ≤ t ≤ t_{4}, ξ(t) = τ(v̄_{4}, \frac{t_{2}−t}{t_{2}−t_{1}} s_{−ē_{3}} + \frac{t−t_{1}}{t_{2}−t_{1}} sē_{4}))(c̄_{4}(t)).

For t_{4} ≤ t ≤ t_{5}, ξ(t) = τ(v̄_{5}, s_{−ē_{4}})(c̄_{5}(t)).

We now only need to see that L(ξ) < L = L(c). The five pieces ξ_{i} of ξ correspond to the pieces c_{i} of c. It is easy to see that L(c_{i}) = L(ξ_{i}) for i = 1, 3, 5 since one comes from the other by an isometry composed with a vertical shift (also an isometry). For i = 2 we note that π_{X_{v_{2}}}(c̄_{2}) is the same as π_{X_{v_{2}}}(ξ_{2}) and hence L(ξ_{2}) < L(c_{2}) since:

|ℝv̄_{2}(ξ_{2}(t_{2})) − ℝv̄_{2}(ξ_{2}(t_{1}))| = |ℝv̄_{2}(c̄_{2}(t_{2})) − ℝv̄_{2}(c̄_{2}(t_{1})) + sē_{2} − s_{−ē_{1}}| <
< |ℝv̄_{2}(c̄_{2}(t_{2})) − ℝv̄_{2}(c̄_{2}(t_{1}))|.

The i = 4 case is similar.

We need now consider the case where N_{v} is of non-oriented type and e is an edge with initial vertex v. This case is similar to but easier than the previous case. We can choose our closed geodesic c to lie in only two Seifert components of N crossing the boundary corresponding to e exactly twice and nearly vertical (in either - up or down - sense). Since N_{v} is of non-oriented type we can choose c so that in the universal cover the corresponding edges ē_{1} and ē_{1} are related by an isometry that reverses the vertical direction and hence sē_{1} = −sē_{2}. An argument similar to above then forces s_{e} = 0.

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