Quasi-Isometries of Graph Manifolds Do Not Preserve Non-Positive Curvature

QUASI-ISOMETRIES OF GRAPH MANIFOLDS DO NOT PRESERVE NON-POSITIVE CURVATURE

DISSERTATION

Presented in Partial Fulﬁllment of the Requirements for the Degree Doctor of

Philosophy in the Graduate School of the Ohio State University

Andrew Nicol, MS

Graduate Program in Mathematics

The Ohio State University

2014

Dissertation Committee:

Jean-Fran¸coisLafont, Advisor

Brian Ahmer

Nathan Broaddus

Michael Davis c Copyright by

Andrew Nicol

2014 ABSTRACT

Continuing the work of Frigerio, Lafont, and Sisto [8], we recall the definition of high dimensional graph manifolds. They are compact, smooth manifolds which decompose into finitely many pieces, each of which is a hyperbolic, non-compact, finite volume manifold of some dimension with toric cusps which has been truncated at the cusps and crossed with an appropriate dimensional torus. This class of manifolds is a generalization of two of the geometries described in Thurston’s geometrization conjecture: H3 and H2 × R. In their monograph, Frigerio, Lafont, and Sisto describe various rigidity results and prove the existence of infinitely many graph manifolds not supporting a locally

CAT(0) metric. Their proof relies on the existence of a cochain having inﬁnite order.

They leave open the question of whether or not there exists pairs of graph manifolds with quasi-isometric fundamental group but where one supports a locally CAT(0) metric while the other cannot. Using a number of facts about bounded cohomology and relative hyperbolicity, I extend their result, showing that there exists a cochain that is not only of inﬁnite order but is also bounded. I also show that this is suﬃcient to construct two graph manifolds with the desired properties.

ii ACKNOWLEDGMENTS

I wish to acknowledge all of the hard work put forth by my advisor, Jean-Fran¸cois

Lafont. His wisdom, insights, and patience were greatly appreciated.

iii VITA

2007 ...... B.A. in Mathematics, The College of Wooster

2011 ...... MA in Mathematics, The Ohio State University

2007-Present ...... Graduate Teaching Associate, The Ohio State University

FIELDS OF STUDY

Major Field: Mathematics

Specialization: Geometric Group Theory

iv TABLE OF CONTENTS

Abstract ...... ii

Acknowledgments ...... iii

Vita ...... iv

List of Figures ...... vii

CHAPTER PAGE

1 Introduction ...... 1

2 Background ...... 5

2.1 Principle S1-Bundles and the Euler Class ...... 5 2.1.1 Universal Bundles ...... 7 2.2 Hyperbolic Manifolds ...... 9 2.3 Quasi-Isometries ...... 10 2.4 Graph Manifolds ...... 12 2.5 Various Cohomology Theories ...... 15 2.5.1 Group Cohomology ...... 16 2.5.2 Bounded Cohomology ...... 18 2.6 A proof of lemma 1.3 ...... 22

3 Relative Hyperbolicity and Isolated Flats ...... 25

4 Exploring some Bounded Cohomology ...... 34

5 Condition (b) ...... 42

6 Completing the Proof ...... 48

APPENDICES

A Various Deﬁnitions of Relatively Hyperbolic Groups ...... 50

v Bibliography ...... 56

vi LIST OF FIGURES

FIGURE PAGE

3.1 One chamber of DAg ...... 30

vii CHAPTER 1

INTRODUCTION

In what follows, we will show the following

Theorem 1.1. There exists inﬁnitely many examples of pairs of graph manifolds with quasi-isometric fundamental groups, with the property that one of them supports a locally CAT(0) metric, but the other one cannot support any locally CAT(0) metric.

Furthermore, these examples arise in every dimension n ≥ 4.

This result was inspired by the following question, posed by Roberto Frigerio,

Jean-Francois Lafont, and Alessandro Sisto in [8]:

Problem 1.2. Is there a pair of irreducible graph manifolds with quasi-isometric fundamental groups, with the property that one of them supports a locally CAT(0) metric, but the other one cannot support any locally CAT(0) metric?

There is already some progress on this question for the classical deﬁnition of graph manifold following the work of Gersten (see deﬁnition 2.15). He looks at the unit tangent bundle X, of a surface Σ of genus g ≥ 2. This is a principal S1-bundle.

Using the Milnor-Wood inequality following [18] and [26], the S1-bundle yields a bounded characteristic class for the central extension Z → π1(X) → π1(Σ) from the long exact sequence of ﬁber bundles. Another result of Gersten [9] tells us that fundamental group of the unit tangent bundle is quasi-isometric to the fundamental 1 group of product Σ × S1. Further arguments can conclude that this example answers the above question. However, this example is restricted to dimension 3. Theorem 1.1 extends this to higher dimensions.

To work towards proving the theorem, we will construct a locally CAT(0) space,

N, with certain properties, and a principle S1-bundle,

1 S → M2 → N with Euler class, β, of inﬁnite order and represented by a bounded cocycle. Then, we

1 q.i. will deﬁne M1 = N × S . By theorem 2.14, M1 ∼ M2. Additionally, M1 will be the product of two locally CAT(0) spaces, hence is locally CAT(0). Meanwhile, since ϕ has inﬁnite order, M2 cannot support a locally CAT(0) metric (lemma 2.18). The result, along with the conditions required to get such manifolds, is stated as follows:

Lemma 1.3. Suppose we can ﬁnd a locally CAT(0) n-manifold N which decomposes as N = A ∪f B, where A and B are both truncated, complete, ﬁnite-volume, non- compact, hyperbolic n-manifolds with toric cusps, glued together via f : ∂A → ∂B so that A ∩ B = ∪T n−1. Suppose also that there exists α ∈ H1(A ∩ B) with

∗ 1 1 a) hαi ∩ i (H (A; Z) ⊕ H (B; Z)) = 0

2 2 b) δ(hαi) ∩ im(Hb (N; Z) → H (N; Z)) 6= ∅

1 δ 2 where δ : H (A ∩ B; Z) → H (N; Z) is the map from the Mayer Vietoris sequence and i∗ is the map induced by the inclusions A ∩ B,→ A and A ∩ B,→ B. Then we can ﬁnd a pair of graph manifolds with quasi-isometric fundamental groups with the property that one of them supports a locally CAT(0) metric, but the other one cannot support any locally CAT(0) metric.

2 The manifold N will come from the double of a suitably chosen manifold A. The existence of this A along with α ∈ H2(∂A) satisfying property (a) from lemma 1.3 is a combination of results from [17] and [8]. All that remains is the fact that the image of the subgroup generated by α in H2(DA) is bounded. This uses a number of diagram

2 2 chases and primarily uses the surjectivity of the map Hb (DA, ∂A) → H (DA, ∂A).

This fact follows from the fact that π1(DA) is hyperbolic relative to π1(∂A), which is a consequence of their action on DAg, a space with isolated ﬂats. In chapter 2 we will provide the necessary deﬁnitions. These will include the background material on graph manifolds, group cohomology, and bounded cohomology of spaces and groups. We will also recall some results from Gersten (2.14) and Frigerio,

Lafont, and Sisto (2.18). Then, we will be able to prove the above lemma.

In chapter 3, we define what is means for a metric space to have isolated flats and we verify that the universal covers of graph manifolds have isolated flats. Then, we use this fact along with some results from [13] and [6] to show that for DA, the double of A, a finite volume, non-compact, hyperbolic manifold with toric cusps, π1(DA) is hyperbolic relative to π1(∂A). In chapter 4, we take a detour and explore part of a paper of Bieri and Eckmann.

We will deﬁne relative bounded cohomology and explore their proof of various isomorphisms between cohomology involving a space X and a (possibly disconnected) subspace Y , and the cohomology involving π1(X) and a subgroup (or possibly union of subgroups) π1(Y ) (Theorem 4.2). Working through the proof, we will see that these isomorphisms also hold for bounded cohomology (Theorem 4.3).

In chapter 5, we return to lemma 1.3 and specialize to the case where A = B, A is formed by truncating A at the cusps, and N = DA, the double of A. We will show that in this setting, if condition (a) is satisﬁed, then condition (b) in the lemma is automatically satisﬁed. As a result, we can restate lemma 1.3 as follows:

3 Lemma 1.4. Let A be a finite volume non-compact hyperbolic manifold, with all cusps diffeomorphic to the product of a torus and [0, ∞). Let A be the compact manifold obtained by truncating the cusps. Suppose we can find a non-trivial cohomology class

1 ∗ 1 1 α ∈ H (A ∩ A) with hαi ∩ i (H (A; Z) ⊕ H (A; Z)) = 0. Then we can ﬁnd a pair of graph manifolds with quasi-isometric fundamental groups with the property that one of them supports a locally CAT(0) metric, but the other one cannot support any locally CAT(0) metric.

Finally, in chapter 6, we answer the above question posed by Frigerio, Lafont, and

Sisto, combining some of their results with our results in section 4, to show that there exist inﬁnitely many examples of such manifolds satisfying condition (a) from lemma

1.3.

4 CHAPTER 2

BACKGROUND

2.1 Principle S1-Bundles and the Euler Class

Deﬁnition 2.1. A ﬁber bundle consists of a (connected) base space B, a total space

E, a projection π : E → B, a space F called the ﬁber such that for any x ∈ B, π−1(x) is homeomorphic to F . Furthermore, for each x ∈ B, there exists Ux ⊂ B containing

−1 x such that π (Ux) is homeomorphic to Ux × F and both of their projections onto

Ux agree. In other words, the following diagram commutes:

−1 ≈ π (Ux) / Ux × F

y Ux For a topological group G, a principal G-bundle has the additional structure of a right G action on the total space E. This right action preserves the ﬁbers in the sense

π(x · g) = π(x) for all x ∈ E and g ∈ G, and the action is free and transitive on these

ﬁbers.

A section of a ﬁber bundle is a map f : B → E such that π ◦ f = idB. The existence of a section is equivalent to a trivialization E ∼= F × B.

The Euler class, which is a certain cohomology class, is the obstruction to the existence of a section. The following discussion can be found in more detail in part

III of [24].

5 p Let E → B be a G-bundle with ﬁber F . The obstruction cocyle, or Euler class, is the obstruction to the existence of a section f : B → E. If L is a subcomplex of B and f is deﬁned on L, then f can gradually be extended to higher dimensions of B.

This extension runs smoothly until dimension q, where πq(F ) is the first non-trivial homotopy group of the fiber F . In this case, f has been extended from being defined on L to being defined on L as well as the p-skeleton of B: f : L ∪ Bp → E. Let σ be an oriented (q + 1)-cell in B, which induces an orientation on ∂σ. Then f|∂σ along −1 ∼ with the orientation on ∂σ determine an element c(f, σ) of πq(p (σ)) = πq(F ).

Thus, this element c = c(f, σ) is a homomorphism from (q + 1)-cells to πq(F ). In

1 the setting of S -bundles, c ∈ Hom(C2(B; Z), Z). Steenrod shows several important properties of this cochain, such as the fact that it is in fact a cocycle. Hence, for

1 2 S -bundles, this cocycle deﬁnes an element of H (B; Z) called the Euler class. This element measures the non-existence of a section, as shown in the following theorem.

Theorem 2.2. A ﬁber bundle F → E → B splits as a product E ∼= F × B if and only if the Euler class is trivial.

Another well known fact, which can also be found in [24] is the following.

Deﬁnition 2.3. Let F → E →π B, and let B0 be an arbitrary topological space with a map f : B0 → B. The induced bundle is the bundle F → E0 → B0 whose total space is the subspace E0 ⊂ B0 × E of pairs (b0, e) with f(b0) = π(e), along with the projection π0 : E0 → B0 given by π0(b0, e) = b0. It also comes with a map fˆ : E0 → E deﬁned by fˆ(b0, e) = e.

? ? ? 0 Keeping with this notation, the map f induces f : H (B; Z) → H (B ; Z) and the Euler class e(B0) of the induced bundle is given by e(B0) = f ?(e(B)).

6 2.1.1 Universal Bundles

Definition 2.4. Let G be a group. The universal principal G-bundle is a specific fiber bundle G → EG → BG. BG is called the classifying space and has the property that every G-bundle over a paracompact space M is the pullback of the universal bundle by some continuous map f : M → BG. Further, if g : M → BG is another map so that the bundles induced by f and g are isomorphic, then f ' g.

From this deﬁnition, it is clear that the universal bundle, if it exists, is unique up to G-equivariant homotopy equivalence. Existence is also guaranteed.

Theorem 2.5. Let G be a topological group. There exists a classifying space BG such that for any space B, the set of maps B → BG are in one-to-one correspondence with

G-bundles over B.

A speciﬁc examples of interest for our purposes is the universal S1-bundle. The universal principal S1-bundle S1 → ES1 → BS1 is given by ES1 ∼= S∞ and BS1 ∼=

∞ CP . p Following Hatcher ([12], example 4D.5 p. 439), given the bundle S1 → ES1 →

BS1 (recall ES1 ∼= S∞ is contractible), then it follows from the long exact sequence of homotopy groups for ﬁber bundles that BS1 is simply connected and we can look at the Gysin sequence.

i 1 The Gysin sequence tells us that H (BS ; Z) = 0 for i = 0, 1 and that ∪e : i 1 i+2 1 2 1 H (BS ; Z) → H (BS ; Z) is an isomorphism, where e ∈ H (BS ; Z) is the Euler ? 1 class. Thus, we see that H (BS ; Z) is the polynomial ring Z[e], generated by the degree two class e.

We can gather further information about the universal S1-bundle using a result from Steenrod.

7 ∼ Theorem 2.6. Let G → EG → BG be a universal G-bundle. Then πi(BG) =

πi−1(G).

1 1 ∼ 1 ∼ In particular, we see that the universal S -bundle has π2(BS ) = π1(S ) = Z 1 and all other homotopy groups are trivial. Thus, BS is a K(Z,2). This leads to the following alternate description, which can be found in [22].

Let X be a nice space (manifold or CW-complex). By the universal coeﬃcient

k theorem, H (K(A, k); A) = Hom(Hk(K(A, k); A),A). However, since K(A, k) has ∼ trivial homotopy groups up to dimension k − 1, it follows that Hk(K(A, k)) =

πk(K(A, k)) = A, so in fact we have

k H (K(A, k); A) = Hom(Hk(K(A, k); A),A) = Hom(A, A).

The identity map id : A → A corresponds to some ι ∈ Hk(K(A, k); A). Thus, for f ∈ [X,K(A, a)], f ∗ : Hk(K(A, k); A) → Hk(X; A), so f ∗(ι) ∈ Hk(X; A). Thus, there is a correspondence Hk(X; A) ∼= [X,K(A, k)] given by f ∗(ι) ↔ f.

1 2 In particular, principal S -bundles over a manifold X are classified by H (X; Z). 1 ∞ 1 So if f : X → BS = CP classifies a principal S -bundle, then the bundle is ∗ 2 classified by e = f (ι) ∈ H (X; Z), defined to be the Euler class of the bundle. The following will be used to construct a graph manifold with desired properties. For completeness, the proof is also provided.

2 Lemma 2.7. Given a space X and an element α ∈ H (X, Z), there exists a principal S1-bundle over X with Euler class α.

This relationship can be seen from the theorem mentioned previously saying that principal S1-bundles over a space X are classiﬁed by homotopy classes of maps of X

1 1 ∞ 1 1 ∞ into BS ,[X,BS ]. We know that CP is a model for BS , so [X,BS ] = [X, CP ]. ∞ ∞ Since CP is a K(Z, 2), so we get [X, CP ] = [X,K(Z, 2)]. Finally, to see how this 2 relates to H (X; Z), we recall a theorem from [12]: 8 Theorem 2.8. There are natural bijections T :[X,K(G, n)] → Hn(X; G) for all

CW complexes X and all n > 0, with G any abelian group. Such a T has the form

T ([f]) = f ∗(α) for a certain distinguised class α ∈ Hn(K(G, n); G).

2.2 Hyperbolic Manifolds

We recall that a manifold is a space where each point locally looks like euclidean space. More concretely, if M is an n-dimensional manifold without boundary, then each point of a manifold M has a neighborhood which is homeomorphic to a subset

n of R . If M has boundary, then for x ∈ ∂M, there is an open neighborhood of x n−1 homeomorphic to a subset of R × R≥0 where x is sent to a point of the form

(u1, . . . , un−1, 0) With the right additional conditions, we can deﬁne a hyperbolic manifold.

n Deﬁnition 2.9. Hyperbolic n-space, H , is the topological space which is a simply connected complete Riemannian manifold with constant curvature -1.

A hyperbolic manifold is a manifold where every point looks locally like hyperbolic space. Similarly to above:

Deﬁnition 2.10. A hyperbolic n-manifold, M n, is a complete Riemannian n-manifold of constant sectional curvature -1.

Let M be a complete finite-volume non-compact hyperbolic manifold with toric cusps. The presence of cusps follow from the noncompactness of M, and these cusps taper off quickly due to the fact that M has finite volume. These cusps support a

n−1 n−1 foliation by closed torus, so each torus is diﬀeomorphic to T × R≥0, where T is the (n − 1)-dimensional torus. Manifolds of this type will be the starting point of the construction of graph manifolds, as we will soon see.

9 2.3 Quasi-Isometries

When viewing metric spaces, often times it is useful to look at the large scale geometric properties of the space. An equivalence relation on metric spaces which preserves this course structure is quasi-isometry. Roughly speaking, if two metric spaces are quasi- isometric, then when viewed from far away, they look identical. We make this precise with the deﬁnition.

Deﬁnition 2.11. Let (X, dX ) and (Y, dY ) be two metric spaces and f : X → Y a not necessarily continuous map between the two spaces. The map f is called a

(λ, )-quasi-isometry if it satisﬁes the following two conditions:

• There exist constants λ ≥ 1 and ≥ 0 such that for every x0, x1 ∈ X, we have the inequality

1 d (x , x ) − ≤ d (f(x ), f(x )) ≤ λd (x , x ) + λ X 0 1 Y 0 1 X 0 1

• There is some constant c ≥ 0 such that for every point y ∈ Y there exists x ∈ X

such that dY (f(x), y) < c. In other words, Y is covered by a c-neighborhood of the image of f.

When such a map f : X → Y exists, we say that X and Y are quasi-isometric.

We can also discuss the notion of groups being quasi-isometric. This is one of the ideas behind geometric group theory: encoding groups as geometric objects. To do this, we look at the Cayley graph.

Definition 2.12. Given a finitely generated group G with finite generating set S, which we will take to be closed under inversion (S = S−1), the Cayley graph of G is formed as follows:

10 • the vertex set V is the set of elements of G.

• an edge connects vertex g0 and g1 if there is an element s ∈ S such that g1 = g0s.

We make each edge have unit length, resulting in a metric space. We say that groups

G0 and G1 are quasi-isometric if their Cayley graphs are quasi-isometric as metric spaces.

Note that this metric is equivalent to the word metric, where d(g0, g1) is the

−1 minimal length of the freely reduced word g0g1 with respect to the generating set S. We also note that while diﬀerent generating sets might produce diﬀerent Cayley graphs, the Cayley graphs will be quasi-isometric.

We also recall the Milnor-Schwarz lemma. This is a very powerful result that connects groups and spaces on which they nicely act. This can be used both directions.

Given a space, we can ask which groups act nicely on the space. On the other hand, given a group, we can ask which spaces does it nicely act on.

Lemma 2.13. (Milnor-Schwarz Lemma) Let X be a proper geodesic metric space and let G be a group acting geometrically on X. Then G is finitely generated. Fur- thermore, for any finite generating set S and x0 ∈ X and giving G the word metric with respect to S, the map G → X defined by g 7→ g.x0 is a quasi-isometry.

We use the following result from Gersten [9] to conclude that our two graph manifolds we construct have quasi-isometric fundamental groups.

Theorem 2.14. Suppose we have a central extension

1 → Z → E → G → 1 whose characteristic class is deﬁned by a 2-cocycle f : G × G → Z taking only a ﬁnite 2 set of values in Z. This is equivalent to the characteristic class in H (G, R) being bounded. In this situation, the group E is quasi-isometric to the product Z × G. 11 1 We will use this result by letting one of the manifolds, M1, be the product S ×DA,

1 where DA is the double of A. The second graph manifold, M2, will be a principal S bundle over DA:

1 S → M2 → DA.

Looking at the long exact sequence of homotopy groups associated with this ﬁber bundle, we get

1 → Z → π1(M2) → π1(DA) → 1.

Along with some assumptions from lemma 1.3, we will be able to conclude that M1 and M2 are quasi-isometric.

2.4 Graph Manifolds

We follow the deﬁnition provided by Frigerio, Lafont, and Sisto in [8]:

Deﬁnition 2.15. (High Dimensional Graph Manifold) A compact, smooth n-manifold

M, n ≥ 3 is called a graph manifold if it can be constructed in the following way:

(1) For i = 1, . . . , r, let Ni be a complete, ﬁnite-volume, non-compact hyperbolic

ni-manifolds Ni with toric cusps, with 3 ≤ ni ≤ n.

(2) Let Ni be the manifold obtained by truncating the cusps of Ni, that is, by re-

moving from Ni a horospherical neighborhood of each cusp.

n−ni k 1 k (3) Let Vi = Ni × T , where T = (S ) is the k-dimensional torus.

(4) Glue together the different Vi’s along various paired boundary components. Each gluing must be via an affine diffeomorphism of the boundary tori and must result

in a connected manifold of dimension n.

12 The submanifolds, Vi, are the pieces of M. We call Ni the base of Vi. Subsets of

n−ni the form {?} × T ⊆ Vi are called ﬁbers of Vi. By this construction, the boundary components of each piece, which we call the walls of M, are either glued to another boundary component or are a subset of ∂M. The boundary tori which are glue together are called internal walls of M, while the other tori which are components of ∂M are called external walls.

Following [8], it is important to note that this definition does not agree with the classical definition of graph manifold in dimension 3. This definition does not allow

2 general finite volume quotients of H × R, nor does the definition allow pieces to be the product of a hyperbolic surface and a circle. However, the definition does allow purely hyperbolic pieces.

This decomposition of graph n-manifold M into pieces V1,...,Vk induces a structure on π1(M) as the fundamental group of a graph of groups GM . Consider a graph with a vertex vi for each piece Vi of M. Associate to this vertex the group π1(Vi).

Each gluing between pieces Vi and Vj corresponds to an edge between the vertices

n−1 vi and vj with associate edge group Z . The inclusion map ∂Vi ,→ M induces n−1 the homomorphism Z → π1(Vi). This gives the structure of the graph of groups

π1(M) = π1(GM ). The manner in which the pieces are glued together, each boundary torus of a piece,

Vi is either a boundary component of Vi or is glued to a boundary component of some

n−1 Vj, in which case it supports a smooth bicollar diﬀeomorphic to T × [−3, 3]. On this smooth bicollar, a function can be deﬁned to transition from a Riemannian metric on one piece to a Riemannian metric on an adjacent piece. This yields Corollary 2.7 from [8]:

Corollary 2.16. Suppose M is a graph manifold and let U ⊆ M be the union of the bicollars of the toric hypersurfaces of M \ ∂M corresponding to the boundary 13 components of the pieces of M. Then M admits a Riemannian metric g which extends the restriction to M \ U of the product metrics originally deﬁned on the pieces of M.

n−ni Continuing with the notation of Vi = N i × T , then Ne i is the complement

ni Bi of an equivariant disjoint collection of open horoballs in H , sometimes called a

n−ni neutered space. Then Vei = Bi × R which embeds in Mf. The universal cover Mf has a structure as described in Corollary 2.13 of [8]. A slight simpliﬁcation of this result follows.

Corollary 2.17. M admits a Riemannian metric such that Mf can be turned into a space such that:

n−n1 (1) The lifts Vei of the pieces Vi are isometric as Riemannian manifolds to Bi×R

(2) If W is an internal wall of M, then the lifts Wf of W are diﬀeomorphic to

n−1 R × [−1, 1].

n−1 (3) If W is an external wall of M, then the lifts Wf of W are diﬀeomorphic to R .

The structure described in this corollary is called a tree of spaces. Details can be found in [8].

A special case of importance is when A is a complete, non-compact, hyperbolic n-manifold with toric cusps. Form A by truncating A at the cusps, resulting in boundary tori A1,A2,...Ar. Then DA, the double of A, is deﬁned to be A glued to 0 another copy A of A along corresponding boundary tori by the identity map. In this case, forming a Riemannian metric that extends from each copy of A to all of M no longer requires a smooth bicollar structure along the boundary tori of ∂A. From this,

n−1 the lifts of the internal walls are just R , a collection of isolated ﬂats, which will be deﬁned later in 3.2.

There is an action of π1(M) on Mf by deck transformations. In the case of M =

DA, π1(A) stabilizes some lift of A and for each conjugate of π1(A), there is a lift 14 of A which it stabilizes. Conversely, for each lift of A, there is a conjugate of π1(A) 0 which stabilizes it. Of course this is also true for A . Similarly, π1(Ai) stabilizes some lift Ai and for each conjugate of π1(Ai), there is another lift of Ai which it stabilizes.

Conversely, for each lift of Ai, there is a conjugate of π1(Ai) which stabilizes it. In their monograph [8], Frigerio, Lafont, and Sisto give other properties in Lemma

2.14. Additionally, they present the following obstruction for certain S1-bundles to support a locally CAT(0) metric.

Lemma 2.18. Let K be a compact topological manifold supporting a locally CAT(0) metric, and let S1 → K0 → K be a principal S1-bundle over K (so that K0 is also compact). If K0 supports a locally CAT(0) metric, then the Euler class e(K0) has

ﬁnite order in H2(K).

We will use the contrapositive of this lemma, which will tell us that if we have a space, K0 as above, whose Euler class is of inﬁnite order, then K0 cannot support a locally CAT(0) metric.

2.5 Various Cohomology Theories

There are several types of cohomology that we will need. We will use standard cohomology of a topolocal space, group cohomology, bounded cohomology of spaces, and bounded group cohomology. Additionally, we will need to know how to deﬁne these cohomology theories for a space relative to a disjoint collection of subspaces and, analogously, cohomology of a group relative to a collection of subgroups. Note that bounded cohomology is not actually a proper cohomology theory. Bounded cohomology does not in general satisfy excision, as L¨ohpoints out in Caveat 2.4.9

[15]. For much of our discussion on cohomology we will be using [15]. We will use the

15 same notation from [15] unless other notation better serves our purposes. We begin by exploring group cohomology.

2.5.1 Group Cohomology

There are multiple ways to deﬁne group cohomology. One way is to deﬁne the cohomology of a group G is by the cohomology of a certain space with fundamental group

G. The space used is called the classifying space, BG.

Deﬁnition 2.19. (Model of BG) If G is a discrete group, then a pointed, connected ∼ CW-complex (X, x) together with an isomorphism π1(X, x) = G is a model of BG if the universal cover of X is contractible. In other words, a model of BG is a K(G, 1).

Deﬁnition 2.20. (Group Cohomology) Given a group G along with a model XG of

∗ ∗ BG, we deﬁne H (G; Z) = H (XG; Z)

Deﬁnition 2.21. Let R be a ring. an R-module A is called projective if it satisﬁes the following lifting property:

For every surjective R-homomorphism π : B → C and every R-homomorphism

α : A → C, there is an R-homomorphism α : A → B such that π ◦ α = α. This is shown as follows

A α α B π / C / 0.

Deﬁnition 2.22. (Resolutions) Let R be a ring and let A be an R-module. A resolution (B∗, ∂∗) of A is an exact sequence (possibly inﬁnite) of R-modules:

∂1 ∂0 ... −→ B1 −→ B0 −→ A → 0

where each Bi is an R-module and is the augmentation map. 16 A projective resolution of A is a chain complex (P∗, ∂∗) of projective R-modules with augmentation map : P0 → A.

This gives us an additional way to describe group cohomology:

? ? Deﬁnition 2.23. If G is a discrete group, then H (G; A) := ExtZG(Z,A), where Z is the trivial G-module.

In other words, take a projective resolution of Z and apply the contravariant functor HomZG(−,A). The group cohomology of G with coeﬃcients in Z is found by computing the homology of this dual complex.

We can use resolutions to give an alternate deﬁnition of group cohomology via the bar resolution.

∗ Deﬁnition 2.24. (The bar resolution, C∗(G), C (G; A)) Let G be a discrete group.

The bar resolution of G is the ZG-chain complex C?(G) deﬁned by M Cn(G) := Z · g0 · [g1|g2| · · · |gn] g∈Gn+1 with G-action:

0 0 g · (g0 · [g1|g2| · · · |gn]) = (g g0) · [g1|g2| · · · |gn]

The diﬀerential for this resolution is given by

Cn(G) → Cn−1(G)

g0 · [g1|g2| · · · |gn] 7→ g0g1 · [g2| · · · |gn] n−1 X j + (−1) g0 · [g1| · · · |gj−1|gjgj+1|gj+2| · · · |gn] j=1

n + (−1) g0 · [g1|g2| · · · |gn−1]

Finally, we can give the deﬁnition with coeﬃcients in a module A to be

? C?(G; A) := C?(G) ⊗G AC (G; A) := HomG(C?(G),A) 17 R C∗ (G) := C∗(G) ⊗Z R

The bar resolution can be used to deﬁne the cohomology of a group as follows:

Deﬁnition 2.25. Let G be a discrete group. The group cohomology of G with coeﬃ-

? ? ? cients in Z is deﬁned by H (G; Z) := H (C (G; Z)).

There is a nice relation between an aspherical space and its fundamental group, which we recall now.

k ∼ k Lemma 2.26. If X is aspherical, then H (X; Z) = H (π1(X); Z).

2.5.2 Bounded Cohomology

We now recall the deﬁnitions of bounded cohomology of a space and bounded cohomology of a group, as presented in [15]. Then we will look at a result analogous to

Lemma 2.26. Note that the deﬁnitions presented here are not as general as those presented in [15], since we will only need to consider the case with Z coeﬃcients. Another great source for this material is [10].

Deﬁnition 2.27. Let X be a pointed, connected CW-complex with Γ = π1(X). Then we deﬁne

? ˜ Cb (X; Z) := bHomΓ(C?(X; R), Z) where bHomΓ(−, Z) are bounded homomorphisms to Z. Then deﬁne the bounded cohomology groups of X with coeﬃcients in A to be

? ? Hb (X; A) := H (Cb(X; A))

From here, we can deﬁne bounded cohomology of groups.

18 Deﬁnition 2.28. Let G be a group and let XG be a model for BG. Then we simply deﬁne

? ? Hb (G; A) := Hb (XG; A).

L¨ohgives another way of deﬁning bounded cohomology of groups from a combinatorial approach, using resolutions.

Definition 2.29. Let G be a discrete group. The Banach bar complex with coefficients in Z is defined by

? R Cb (G; Z) := bHom(C? (G), Z),

R where C? (G) was deﬁned in 2.24. We can then take the homology of this complex to get bounded group cohomology:

? ? Hb (G; Z) := H (Cb(G; Z))

There is a nice result regarding the bounded cohomology of amenable groups. We recall that a discrete groups G is called amenable if there is a ﬁnitely additive, left- invariant probability meanure on it. Some examples include ﬁnite groups, amenable groups, and solvable groups.

Noskov characterized amenable groups by vanishing bounded cohomology groups in ([21], p. 1068). The following comes from Theorem 8 of [14].

Theorem 2.30. Let G be a discrete group. Then the following are equivalent:

(a) G is amenable.

1 (b) Hb (G; Z) = 0.

k (c) For all k ∈ N>0 we have Hb (G; Z) = 0.

19 We now arrive at an extremely useful tool for bounded cohomology, which says that the bounded cohomology of a space depends only on the fundamental group of the space.

k ∼ k Lemma 2.31. Hb (X; Z) = Hb (π1(X); Z)

In particular, if two spaces, X and Y , have the same fundamental group, then we k ∼ k can apply the above lemma to conclude that Hb (X) = Hb (Y ) for all k. Additionally, we see that in the deﬁnition of bounded cohomology of a group G found in deﬁnition

2.28, the choice of using the model space XG was unnecessary. In fact, any space with fundamental group G could be used.

Lemma 2.31 combines with lemma 2.26 in the following way:

Lemma 2.32. If X is aspherical, then the following diagram commutes:

∼ 2 = 2 Hb (X; Z) / Hb (π1(X); Z)

∼ 2 = 2 H (X; Z) / H (π1(X); Z) Proof. To see the commutativity of this diagram, we must ﬁrst understand the isomorphisms. The isomorphisms are induced by maps on the level of the chain complexes.

Thus, we will show commutativity on the level of cochain complexes, following the construction given in [15].

The idea is to construct maps

˜ R ϕ? : C?(X; R) → C? (Γ)

R ˜ ψ? : C? (Γ) → C?(X; R) where Γ = π1(X) and with the properties that:

R · the composition ϕ? ◦ ψ? is homotopic to the identity map on C? (Γ).

˜ · the composition ψ? ◦ ϕ? is homotopic to the identity map on C?(X; R) 20 · there exist corresponding RΓ-chain homotopies that consist of bounded linear maps in every degree.

These details are shown in [15]. L¨ohalso shows that these maps are norm non- increasing, with respect to the corresponding `1-norms, making the isomorphism an isometric isomorphism, but this is a fact we do not need for our purposes. We now consider the following diagram ψ ˜ 2 R C2(X; R) ←−−− C2 (Γ)

ψ ˜ 2 R C2(X; R) ←−−− C2 (Γ) which is clearly still commutative. Recall bHom(A, Z) is the set of homomorphisms from the set A to Z with bounded image, and that bHom(−, Z) is a contravariant functor. Applying the functors bHom(−, Z) to the top row and Hom(−, Z) to the bottom row induces the following:

˜ R bHom(C2(X; R), Z) / bHom(C2 (Γ), Z)

˜ R Hom(C2(X; R), Z) / Hom(C2 (Γ), Z) which is still commutative. Finally, taking homology yields the commutative diagram:

2 2 Hb (X; Z) / Hb (Γ; Z)

H2(X; Z) / H2(Γ; Z) This completes the proof of the commutativity of the diagram.

21 2.6 A proof of lemma 1.3

At this point, we return our attention to lemma 1.3. We will see how the assumptions in the lemma are suﬃcient to answer our questions. This proof uses ideas from Propo- sition 12.2 from Frigerio, Lafont, and Sisto’s paper. We will see how we can expand on these ideas using the added assumption from condition (b) and the previously mentioned facts about bounded cohomology to work towards Theorem 1.1.

Proof. Assume there exists such an N. First, we will construct two spaces, M1 and

M2. Then we will show that they have quasi-isometric fundamental groups. Next, we will see that M1 can support a locally CAT(0) metric and M2 cannot. Finally, we will verify that they are indeed graph manifolds.

1 By Lemma 2.7, there exists a principal S -bundle M2 over N,

1 S → M2 → N

2 1 with Euler class e(M2) = β ∈ H (N; Z). Now we set M1 = N × S . q.i. Claim: π1(M1) ∼ π1(M2) Passing to the long exact homotopy sequence for ﬁber bundles, we get the following exact sequence

1 1 ... → π2(N) → π1(S ) → π1(M2) → π1(N) → π0(S ) → ...

From here, we note that since N is aspherical, all of its higher homotopy groups

1 1 vanish. We also know that the only non-trivial homotopy group of S is π1(S ) = Z. Thus, our long exact sequence becomes the short exact sequence

1 → Z → π1(M2) → π1(N) → 1

Using Theorem 2.14, if we show that this central extension has bounded charac-

q.i. teristic class, we will be able to conclude that π1(M2) ∼ Z × π1(N) = π1(M1), as desired. 22 Since N is locally CAT(0), it is aspherical, so lemma 2.32 applies. Set β = δ(αk) ∈

2 k H (N; Z), where k is the smallest positive integer such that α lies in the intersection 2 (see condition (b)). Then there exists β ∈ Hb (N; Z) which maps to β under the 2 comparison map. Then, by the isomorphisms, there exists γ ∈ Hb (π1(N); Z) and 2 by the commutativity of the diagram, there exists γ ∈ H (π1(N); Z) ﬁtting into the diagram as follows:

2 2 ' β Hb (N) / Hb (π1(N)) γ

2 2 β H (N) / H (π1(N)) 7 γ

2 Let f be a cocycle representative from the homology class γ ∈ H (π1(N); Z) with the property that f takes only ﬁnitely many values in Z. Such a cocycle, f, exists 2 2 because γ is in the image of the comparison map Hb (π1(N); Z) → H (π1(N); Z). Thus, we can apply theorem 2.14 to conclude that

q.i. π1(M2) ∼ Z × π1(N) = π1(M1).

Thus, M1 and M2 have quasi-isometric fundamental groups.

We now show that M1 supports a locally CAT(0) metric while M2 cannot support

1 a locally CAT(0) metric. Since M1 = N × S is a product of locally CAT(0) spaces,

M1 supports a locally CAT(0) metric. On the other hand, M2 cannot support a

2 locally CAT(0) metric. We can see this by ﬁrst noting that β = e(M2) ∈ H (N; Z) has inﬁnite order. This follows from hypothesis (a) and the fact that β = δ(α).

Hypothesis (a), tells us that no power of α is in the image of i∗ : H1(A) ⊕ H1(B) →

H1(A ∩ B), hence, no power of α is in the kernel of δ : H1(A ∩ B) → H2(N).

Therefore, the image of α under δ is also of inﬁnite order. Thus, lemma 2.18 tells us that M2 cannot support a locally CAT(0) metric.

23 Finally, we verify that M1 and M2 are indeed graph manifolds. M1 is clearly a graph manifold, bring the product of the graph manifold N and S1. We now verify that M2 is a graph manifold. To do this, we will follow the ideas in the proof of [8]

1 12.2: We look at the ﬁber bunder S → M2 → N = A ∪ B and observe that M2 decomposes as a union M2 = MA ∪ MB, where MA is the preimage of A and MB is

1 the preimage of B. We will look at the Euler class of the bundle S → MA → A.

? 2 2 The inclusion map iA : A,→ N induces a map iA : H (N; Z) → H (A; Z). Similarly, ? 2 2 ? 2 2 2 iB : H (N; Z) → H (B; Z). Then the map i = iA −iB : H (N; Z) → H (A)⊕H (B) is the map that occurs in the Mayer-Vietoris sequence:

? 1 δ 2 i 2 2 · · · → H (A ∩ B; Z) → H (N; Z) → H (A; Z) ⊕ H (B; Z) → · · ·

but then i?(f) = i?(δ(α)) = 0 by exactness of the sequence. Similarly, the Euler

2 class of the MB bundle also has trivial image in H (B; Z). By Theorem 2.2, this tells 1 1 us that MA is homeomorphic to S × A and MB is homeomorphic to S × B. As before, this tells us that MA and MB are pieces of a graph manifold. We endow each piece with the smooth structure induced by the product of smooth manifolds. All that remains to verify is that the gluings between MA and MB are affine diffeomorphisms. If they are not, we can replace it by a homotopic affine diffeomorphism without

1 aﬀecting the Euler class of the corresponding principal S -bundle, and hence, M2 is a graph manifold. This proves the lemma.

24 CHAPTER 3

RELATIVE HYPERBOLICITY AND ISOLATED FLATS

We are now in a position to show that, given a truncated complete, finite-volume, non-compact hyperbolic n-manifold with toric cusps, then its fundamental group is hyperbolic relative to its cusp subgroups. To work towards this, we beging by defining what it means for a group to be hyperbolic relative to a finite collection of subgroups.

Then we will follow the work of Hruska and Kleiner [13] as well as Drut¸u and Sapir

[6].

The notion of a group being hyperbolic relative to a subgroup was originally presented by Gromov in [11]. In [2], Bowditch gave a combinatorial formulation and showed it was equivalent to Gromov’s original definition. In [7], Farb gives an alternate definition using coned-off Cayley graphs. In [25], Szczepanski showed that these two definitions are not equivalent, and that Farb’s definition is weaker. He shows that if a group G is hyperbolic relative to a subgroup H by Gromov’s definition, then

G is hyperbolic relative to H by Farb’s deﬁnition, but not conversely. More details can be found in section A. We now present the deﬁnition which follows from Gromov, as presented in [20].

0 Deﬁnition 3.1. Let Γ be a group and Γ = {Γi|i ∈ I} a family of subgroups of Γ. Γ is hyperbolic relative to Γ0 if there exists a graph K with vertex set V on which Γ acts with the following properties:

• Γ is finitely generated. 25 • I is finite, Γi is finitely generated for each i.

• K is ﬁne and has thin triangles.

• There are ﬁnitely many orbits of edges and each edge stabilizer is ﬁnite.

0 0 • There exists a Γ-invariant subset V such that V∞ ⊆ V ⊆ V and the stabilizers

0 in V are precisely Γi and their conjugates.

Where V∞ is the set of vertices of inﬁnite valence.

We now set some notation for remainder of this section. Let A be a complete,

ﬁnite-volume, non-compact hyperbolic n-manifold with toric cusps, and let A be the result of truncating A at the cusps. We then have that the boundary of A is disjoint collection of tori, which we will label A1,...,Ar. DA is the double of A, formed by taking two copies of A and gluing them together via the identity map along their corresponding boundary components. Let Γ = π1(DA) and Γi = π1(Ai) be the cusp

n−1 subgroups. Since these cusps are all tori, each Γi is isomorphic to a copy of Z , each of which injects into Γ [8]. Set X = DAg, the universal cover of DA, which has ˜ ˜ subsets A = {Aij}, where Aij is the collection of lifts of Ai, i = 1, . . . , r, j ∈ J for some countable index set J.

From [13],

Definition 3.2. A k-flat is defined to be an isometrically embedded copy of Euclidean

k space E for k ≥ 2. Define Flat(X) to be the space of all flats in X with the topology of Hausdorff convergence on bounded sets. A CAT(0) space X with geometric group action has isolated flats if it has an equivariant collection F of flats which is closed and isolated in Flat(X) and each flat F ⊂ X is contained in a uniformly bounded tubular neighborhood of some F 0 ∈ F.

26 In particular, X = DAg has isolated ﬂats A, consisting of the collections of lifts of ˜ n−1 the gluing tori. Each Aij is diﬀeomorphic to R × [−1, 1]. Also, the lifts of A are neutered spaces, which is hyperbolic space with a collection of horoballs removed. A more detailed description of the universal cover is discussed in [8].

We wish to draw a conclusion about Γ being hyperbolic relative to some collection of subgroups. Clearly, Γ acts geometrically on X, so we can follow Theorem 1.2.1 from [13], shown here.

Theorem 3.3. Let X be a CAT(0) space and Γ a group acting geometrically on X.

The following are equivalent.

(1) X has isolated ﬂats.

(2) Each component of the Tits boundary ∂T X is either an isolated point of a standard Euclidean sphere.

(3) X is a relatively hyperbolic space with respect to a family of ﬂats F.

(4) Γ is a relatively hyperbolic group with respect to a collection of virtually abelian

subgroups of rank at least two.

To more fully understand property 4 above in our case of Γ = π1(DA) acting on

X = DAg, which satisﬁed property 1, we explore the proof of (1) ⇒ (3) ⇒ (4). We make note of corollary 3.1.3 from [13]:

Corollary 3.4. The set of stabilizers of ﬂats F ∈ F is precisely the set A os maximal virtually abelian subgroups of Γ with rank at least two.

We wish to explore this proof in more detail, making use of a paper by Drut¸u and

Sapir ([6]), in which they explore quasi-actions by quasi-isometries of a group G on a space X.

First, we make use of Theorem 3.3.6 from [13]. 27 Theorem 3.5. Let X be a CAT(0) space admitting a proper, cocompact, isometric action of a group Γ. If X has isolated ﬂats with respect to a family of ﬂats F, then

X is relatively hyperbolic with respect to F.

So in particular, X = DAg is relatively hyperbolic with respect to A, the lifts of the cusps. While [13] uses the terminology “relatively hyperbolic” for metric spaces, while Drut¸u and Sapir use the terminology “asymptotically tree-graded.” Thus, X is asymptotically tree-graded with respect to A. We will show that the group Γ is asymptotically tree-graded with respect to a speciﬁc family of subgroups. The deﬁnition of this comes from [6].

Definition 3.6. We say that a finitely generated groups Γ is asymptotically tree- graded with respect to the family of subgroups {H1,H2,...,Hk} if the Cayley graph Cayley(G) with respect to some (and hence every) finite set of generators is asymptotically tree-graded with respect to the collection of left cosets {gHi | g ∈ G, i = 1, 2, . . . , k}.

We now turn to the paper from Drut¸u and Sapir [6], where they present the following theorem.

Theorem 3.7. Let X be a space that is asymptotically tree-graded with respect to a collection of subset A. Assume that

(1) A is uniformly asymptotically without cut-points, where all sets A ∈ A are

endowed with the metric induced from X;

(2) every A ∈ A has inﬁnite diameter;

(3) For a ﬁxed x0 ∈ X and every R > 0 the ball B(x0,R) intersects ﬁnitely many A ∈ A.

28 Let G be a ﬁnitely generated group which is quasi-isometric to X. Then there exists subsets A1,...,Am ∈ A and subgroups H1,...,Hm of G such that

(I) every A ∈ A is quasi-isometric to Ai for some i ∈ {1, . . . , m};

(II) Hi is quasi-isometric to Ai for every i ∈ {1, 2, . . . , m};

(III) G is asymptotically tree-graded with respect to the family of subgroups {H1,...,Hm}.

We now verify that the space DAg along with the collection of lifts of the cusps of DA satisfy the conditions above. Since there are ﬁnitely many isometry classes of spaces in the lifts of the cusps, we must verify that each class, is asymptotically without cut-points [6]. In fact, there is only one isometry class in the space of the lifts of the cusps. This means we must show that each asymptotic cone is without

n−1 cut-points. Since the lifts of these cusps are diﬀeomorphic to R × [−1, 1] [8], their asymptotic cones are isometric to Euclidean space, hence they have no cut-points.

n−1 Therefore, condition (1) is satisfied. Also, these copies of R × [−1, 1] have infinite diameter, so condition (2) is satisfied. Finally, condition (3) is satisfied, which can be seen when considering the universal cover DAg described as a tree of spaces [8]. We can see from one of the chambers in figure 3.1.

As R increases, B(x0,R) will intersect more and more horospheres, which are the lifts of the cusps, and even extended into the adjacent chambers, but still will only intersect ﬁnitely many horospheres. Futher discussion on properties of truncated hyperbolic space, also called neutered spaces, can be found in [3] pp.362-366. In

n particular, every disjoint collection of horoballs in H is locally finite. Thus, only n finitely many horoballs meet any compact subset of H . This fact, combined with the fact that there is a lower bound on the distance between these horoballs, give us the fact that B(x0,R) intersects only finitely many lifts of gluing tori.

29 Figure 3.1: One chamber of DAg

Now, we have veriﬁed that the space DAg along with the collection of lifts of the cusps of DA satisfy the conditions above. The group Γ = π1(DA) acts geometrically on DAg, so by Lemma 2.13, Γ is quasi-isometric to DAg, hence, Γ has property (III). Thus, we have that Γ is asymptotically tree-graded with respect to some family of subgroups {H1,...,Hm} of G, where each Hi is quasi-isometric to the lift of some cusp, and each lift of a cusp is quasi-isometric to some Hi. To understand these subgroups better, we use some ideas in the proof of Theorem 5.10 in [6].

First, we recall some information about the universal cover DAg and how Γ acts on

30 it. Γ acts on DAg by isometries via deck transformations. This action Γ × DAg → DAg will be denoted by (g, x) 7→ gx. The stabilizer of a ﬂat A ∈ A is deﬁned by

St(A) = {g ∈ Γ | gA = A}

It is clearly seen that St(A) is a subgroup of Γ. In fact, from previous remarks on the structure of DAg and how π1(DA) acts on it, St(A) is exactly the (possibly conjugate of the) fundamental group of a boundary cusp π1(Ai) of A.

Recall from [8], π1(A) stabilizes a lift of A (for each lift of A, there is a conjugate ˜ of π1(A) which stabilizes it). Similarly, π1(Ai) stabilizes a lift Aij of Ai for some j ∈ J, and for every lift of Ai, there is some conjugate of π1(Ai) which stabilizes it. Additionally, part of Theorem 1.2.2 from [13] is

Theorem 3.8. Let Γ act geometrically on a CAT(0) space X with isolated ﬂats.

Then X and Γ have the following properties.

(1) Quasi-isometries of X map maximal ﬂats to maximal ﬂats.

For properties (2)-(5), see [13]. In particular, the action of Γ on DAg maps ﬂats to ﬂats.

Let q :Γ → DAg and q : DAg → Γ be (L0,C0)-quasi-isometries such that q ◦ q and q ◦ q are at distance C0 from the identity map on DAg and Γ respectively. Finally, we also recall Lemma 5.17 from [6].

Lemma 3.9. For every i ∈ {1, 2, . . . , r} we have hdist(q(Aij), StD(Aij)) ≤ κ, where

κ is a constant depending on L0, C0, and dist(q(1), x0).

Note that StD(Aij) = {g ∈ Γ|hdist(Aij, gAij) ≤ D}. Returning to the proof of Theorem 5.10 from [6], we will see that for our purposes, the proof shows that Γ is hyperbolic relative to π1(Ai), the cusp subgroups. In

31 particular, we will show that Γ is hyperbolic relative to the stabilizers of some of the lifts of the cusps, which will be π1(Ai).

Begin by taking a ﬁnite collection of the lifts of the cusps that intersect B(x0,R) for some radius R, which will be discussed further shortly. We know that π1(Ai) stabilizes some lift Bi of {Aij | j ∈ J}, so take R big enough so that B(x0,R) intersects each Bi for i = 1, . . . , r. Let F = {C1,C2,...,Ck} denote the set of all

flats of DAg that intersect B(x0,R) non-trivially. Let B be the minimal subset of F such that B contains a representative from each Γ-orbit of the flats in F. The set B = {B1,B2,...,Br} satisfies these conditions. In other words, the set B =

{B1,B2,...,Br} is a minimal subset of ﬂats of F such that for each A ∈ F, there exists g ∈ Γ and B ∈ B such that gB = A. Thus, the set B satisﬁes condition (I) of the theorem. An explanation can be found following corollary 5.14 of [6].

Let Hi = St(Bi) = π1(Ai) for i = 1, 2, . . . , r. From this we see that Hi is quasi- isometric to Bi for every i ∈ {1, 2, . . . , r}, fulﬁlling condition (II) of the theorem. All that remains is to show that Γ = π1(DA) is asymptotically tree-graded with respect to this family of subgroups {H1,H2,...,Hr}.

To show that G is asymptotically tree-graded with respect to {H1,H2,...,Hr}, we must show that Cayley(G) is asymptotically tree-graded with respect to {gHi | g ∈ Γ, i = 1, 2, . . . , r}. Using Theorem 5.1 in [6], we know that Cayley(G) is asymptotically tree-graded with respect to {q(A) | A ∈ A}. Therefore, we only need to verify that the sets {gHi | g ∈ Γ, i = 1, 2, . . . , r} and {q(A) | A ∈ A} are at ﬁnite Hausdorﬀ distance from each other. If they are, then their image in any asymptotic cone will coincide.

The fact that the sets {gHi | g ∈ Γ, i = 1, 2, . . . , r} and {q(A) | A ∈ A} are at finite Hausdorff distance from each other follows from how B was defined. For any

A ∈ A, there exists g ∈ Γ and Bi ∈ B such that A = gBi. Using this, q(A) = gq(Bi),

32 which is at uniformly bounded Hausdorﬀ distance from gSt(Bi) = gHi. Thus, {q(A) |

A ∈ A} is contained in a ﬁnite neighborhood of {gHi | g ∈ Γ, i = 1, 2, . . . , r}.

On the other hand, to see that {gHi | g ∈ Γ, i = 1, 2, . . . , r} is contained in a ﬁnite neighborhood of {q(A) | A ∈ A}, let gHi be an arbitrary element of

{gHi | g ∈ Γ, i = 1, 2, . . . , r}. Since Bi is a ﬂat, gBi is also a ﬂat in DAg, hence,

A = gBi for some A ∈ A. Thus, q(A) = gq(Bi), which is at uniformly bounded

Hausdorff distance from gSt(Bi) = gHi. Therefore, {gHi | g ∈ Γ, i = 1, 2, . . . , r} is contained in a finite neighborhood of {q(A) | A ∈ A}, so the two sets are at finite

Hausdorﬀ distance from each other. This means that their image in any asymptotic cone is the same, so Cayley(G) is asymptotically tree-graded with respect to

{gHi | g ∈ Γ, i = 1, 2, . . . , r} and hence, G is asymptotically tree-graded with respect to {H1,H2,...,Hr} = {π1(Ai) | i = 1, 2, . . . r}. Finally, we use one last result from [6].

Theorem 3.10. A ﬁnitely generated group G is asymptotically tree-graded with respect to subgroups {H1,...,Hm} if and only if G is hyperbolic (in the sense of Gromov) relative to {H1,...,Hm} and each Hi is ﬁnitely generated.

Therefore, we are ﬁnally able to conclude the following

Lemma 3.11. Given A a non-compact, ﬁnite volume, hyperbolic manifold with toric cusps, then Γ = π1(DA) is hyperbolic relative to the collection of subgroups Γi =

π1(Ai), where the Ai are the boundary components of A.

33 CHAPTER 4

EXPLORING SOME BOUNDED COHOMOLOGY

Before we explore condition (b) from lemma 1.3, we will need some results of bounded cohomology. We turn to a paper by Bieri and Eckmann. The following notation is a combination of notation from the paper of Bieri and Eckmann’s [1] as well as Mineyev and Yaman’s [20].

0 Let Γ be a group and Γ = {Γi | i ∈ I} a collection of subgroups for some index set I. Then IΓ = ti∈I iΓ/Γi is the disjoint union of copies of Γ for each subgroup in

0 0 0 Γ . Define the quotient Γ/Γ = ti∈I iΓ/Γi, so then ZΓ/Γ = ⊕i∈I Z[iΓ/Γi]. The map 0 : ZΓ/Γ → Z can be defined by sending each generator to 1. The kernel of this map shall be denoted by ∆. Let A be some arbitrary coefficient Γ-module.

00 0 Now, let P be a projective resolution of Z and deﬁne P = ZΓ/Γ ⊗P a projective 0 resolution of ZΓ/Γ . These projective resolution ﬁt into the following diagram

00 00 00 0 ··· / P2 / P1 / P0 / ZΓ/Γ ··· / P2 / P1 / P0 / Z

0 where the vertical maps are projections β : ZΓ/Γ ⊗P → P onto the second factor. These maps are all surjective, with kernel ∆ ⊗ P . Deﬁne P 0 = ∆ ⊗ P the kernel of

34 β : P 00 → P , β induced by . This is a projective resolution of ∆, which expands the above diagram to the following

0 0 0 ··· / P2 / P1 / P0 / ∆

00 00 00 0 ··· / P2 / P1 / P0 / ZΓ/Γ ··· / P2 / P1 / P0 / Z 0 where the upper vertical maps α : ∆ ⊗ P → ZΓ/Γ ⊗ P are inclusion. It is mentioned in [1] that Hk(Γ, Γ0; A) = Hk−1(Γ; Hom(∆,A)) = Extk−1( , Hom(∆,A)). ZΓ Z Furthermore, since ∆ is -free, it follows that Hk(Γ, Γ0; A) ∼ Extk−1(∆,A). Addi- Z = ZΓ Y Y tionally, Extk−1( Γ/Γ0,A) ∼= Extk−1( Γ/Γ ,A) ∼= Extk−1( ,A) ∼= Hk(Γ ; A) ∼= ZΓ Z ZΓ Z i ZΓi Z i i i Hk(Γ0; A). Using these facts with the above projective resolutions, one gets the long exact sequence

· · · → Hk(Γ; A) → Hk(Γ0; A) → Hk+1(Γ, Γ0; A) → Hk+1(Γ; A) → · · ·

γ Now, deﬁne another short exact sequence of resolutions P 00 →i E → ΣP 0 as follows.

0 0 0 0 0 ΣP = P [−1] is P shifted by one degree, ie ΣPn = Pn−1. The mapping cone of α,

0 00 denoted by E = E(α), is deﬁned by En = Pn−1 ⊕ Pn with boundary map

0 00 0 0 00 ∂E(p , p ) = (−∂P 0 (p ), α(p ) + ∂P 00 (p )).

0 00 00 It is a straightforward calculation to show that ∂E∂E(p , p ) = 0. The map i : P → E is the inclusion map, while γ : E → Σ is the projection map γ(p0, p00) = p0. This ﬁts in to the following diagram

00 00 00 ··· / P2 / P1 / P0

··· / E2 / E1 / E0

0 0 ··· / ΣP2 / ΣP1 35 Continuing the discussion in [1], deﬁne the map ϕ : E → P by ϕ(p0, p00) = βp00.

This map commutes with the boundary map ∂.

From the short exact sequences of chain complexes, we obtain two long exact sequences of cohomology groups that ﬁt into the following diagram:

δ β∗ α∗ ··· / Hk−1(P 0; A) / Hk(P ; A) / Hk(P 00; A) / Hk(P 0; A) / ···

∗ = (−1)k+1 ϕ = (−1)k+1 = ··· / Hk(ΣP 0; A) / Hk(E; A) / Hk(P 00; A) / Hk+1(ΣP 0; A) / ··· γ∗ i∗ δ

We note that in the process of obtaining these long exact sequences, we just as easily could have applied the bHom functor instead of the Hom functor before taking the homology. The result would yield two similar long exact sequences of bounded cohomology groups. The discussion in the remainder of this section applies to the current setting as well as the with bounded cohomology.

We wish to show that the above diagram commutes exactly in the second square and up to sign (−1)k+1 in the ﬁrst and last squares.

Proof. We ﬁrst show commutativity of the ﬁrst square, up to sign (−1)k+1. Let

0 k+1 ∗ ∗ k f ∈ Hom(Pk−1,A). We will see that (−1) ϕ δf = γ f in H (E; A). This it to say they diﬀer by a boundary in Hom(Ek,A). So we want to show that there exists

∗ k+1 ∗ Ψ ∈ Hom(Ek−1,A) with δEΨ = γ f − (−1) ϕ δf. To show this, we must ﬁrst make sense of the connecting map δ : Hk−1(P 0; A) →

Hk(P ; A) so that we can understand δf. Looking at the short exact sequence of chain complexes, 0 0 · · · ←−−− Hom(Pk,A) ←−−− Hom(Pk−1,A) ←−−− · · · x x  ∗  ∗ α α

00 ∂P 00 00 · · · ←−−− Hom(Pk ,A) ←−−− Hom(Pk−1,A) ←−−− · · · x x β∗ β∗  

· · · ←−−− Hom(Pk,A) ←−−− Hom(Pk−1,A) ←−−− · · · 36 00 there exists some g ∈ Hom(Pk−1; A) such that

α∗g = f.

Then δf ∈ Hom(Pk; A) is the element such that

∗ β δf = ∂P 00 g.

We note how δP 00 acts on a k-chain h:

δh = (−1)k+1h∂, so in particular, since g is a (k − 1)-chain, we have δg = (−1)kg∂. Now, let (p0, p00) ∈

0 00 Ek = Pk−1 ⊕ Pk be an arbitrary element. Then we have

(γ∗f − (−1)k+1ϕ∗δf)(p0, p00) = γ∗f(p0, p00) − (−1)k+1ϕ∗δf(p0, p00)

= f ◦ γ(p0, p00) − (−1)k+1δf ◦ ϕ(p0, p00)

= f(p0) − (−1)k+1δf(βp00)

= f(p0) − (−1)k+1(β∗δf)(p00)

∗ 0 k+1 00 = (α g)(p ) − (−1) (δP 00 g)(p )

0 k+1 k 00 = g(α(p )) − (−1) (−1) g(∂P 00 p )

0 00 = g(α(p )) + g(∂P 00 p ).

In order to show that this represents the zero cocyle in Hk(E; A), we must ﬁnd

k−1 0 00 an element Ψ ∈ H (E; A) such that ∂EΨ = g(α(p )) + g(∂P 00 p ). Since, Ψ ∈

0 00 0 00 Hom(Ek−1,A) = Hom(Pk−2 ⊕ Pk−1,A), we can decompose Ψ as a sum, Ψ = Ψ + Ψ ,

0 0 00 00 0 where Ψ ∈ Hom(Pk−2,A) and Ψ ∈ Hom(Pk−1,A). Claim: The choice of Ψ = 0 and Ψ00 = g gives us the desired cocycle. We see this claim by taking the coboundary

0 00 0 0 of Ψ = 0 + g and seeing how it acts on (p , p ) ∈ Pk−1 ⊕ Pk:

37 0 00 0 00 δEΨ(p , p ) = Ψ ◦ ∂E(p , p )

0 0 00 = Ψ(−∂P 0 p , αp + ∂P 00 p )

0 0 00 = 0(−∂P 0 p ) + g(αp + ∂P 00 p )

0 00 = g(α(p )) + g(∂P 00 p ),

As desired.

We move on to check commutativity of the second square. Let f ∈ Hk(P ; A). We

∗ ∗ ∗ k 00 00 00 must show that β f = i ϕ f ∈ H (P ; A). For any p ∈ Pk we have,

β∗f(p00) = f(β(p00))

= f ◦ ϕ(0, p00)

= f ◦ ϕ ◦ i(p00)

= (i∗ϕ∗f)(p00).

Thus, β∗f = i∗ϕ∗f ∈ Hk(P 00; A). Since f was arbitrary, the middle square commutes.

We now show commutativity, up to sign (−1)k+1, of the third square. Let

k 00 k 00 ∗ k+1 f ∈ H (P ; A) = H (Hom(P∗ ,A)). We must show that α f = (−1) δf ∈

k+1 0 k+1 0 H (ΣP ; A) = H (Hom(ΣP∗+1,A)). First, we look at the connecting map δ : Hk(P 00; A) → Hk+1(ΣP 0; A). It comes from the short exact sequence of chain complexes: 00 00 · · · ←−−− Hom(Pk+1,A) ←−−− Hom(Pk ,A) ←−−− · · · x x  ∗  ∗ i i

∂E · · · ←−−− Hom(Ek+1,A) ←−−− Hom(Ek,A) ←−−− · · · x x γ∗ γ∗   0 0 · · · ←−−− Hom(ΣPk+1,A) ←−−− Hom(ΣPk,A) ←−−− · · ·

38 ∗ If we let g ∈ Hom(Ek,A) be an element in the preimage of f under the map i :

00 Hom(Ek,A) → Hom(Pk ,A), then we get

i∗g = f.

0 Then δf ∈ Hom(ΣPk+1,A) has the property that

∗ γ δf = δEg

We will see that α∗f − (−1)k+1δf = 0 ∈ Hk+1(ΣP 0,A) by showing that there exists

0 ∗ k+1 0 0 some Ψ ∈ Hom(ΣPk,A), such that (α f − (−1) δf)(p ) = (δΣΨ)(p ) for every

0 0 0 p ∈ ΣPk+1 = Pk. First, we see that

(α∗f − (−1)k+1δf)(p0) = (α∗f)(p0) − (−1)k+1(δf)(p0)

= f(α(p0)) − (−1)k+1(δf)(γ(p0, 0))

= (i∗g)(α(p0)) − (−1)k+1(γ∗δf)(p0, 0)

0 k+1 0 = g(i(α(p ))) − (−1) (δEg)(p , 0)

0 k+1 k+1 0 = g(0, α(p )) − (−1) (−1) g ◦ ∂E(p , 0)

0 0 0 = g(0, α(p )) − g(−∂P 0 p , αp + ∂P 00 0).

0 00 0 00 As before, since g ∈ Hom(Ek,A) = Hom(Pk−1⊕Pk ,A), we can decompose g = g +g ,

0 0 00 0 with g ∈ Hom(Pk−1,A) and g ∈ Hom(Pk,A).

∗ k+1 0 0 00 0 0 0 00 0 00 (α f − (−1) δf)(p ) = g (0) + g (α(p )) + g (∂P 0 p ) − g (α(p )) − g (∂P 00 0)

0 0 = g (∂P 0 p )

k+1 0 0 = (−1) (δP 0 g )(p ).

This is independent of p0, so as desired, α∗f and (−1)k+1δf diﬀer by coboundary, telling us that they represent the same cohomology class in Hk+1(ΣP 0; A). 39 This completes the proof of the commutativity of the diagram.

This commutativity applies nicely for any short exact sequence of long exact sequences, D,→ C Q, where

0 · D is a resolution of ZΓ/Γ ,

· C is a resolution of Z, and

0 · D,→ C induces : ZΓ/Γ → Z.

By exactness and the commutativity, it follows that Q is a resolution of ∆ with dimension shift, so that H0(Q) = 0 and H1(Q) = ∆.

Using the argument above, we can see that D,→ C Q is chain homotopy 00 0 equivalent to P ,→ E ΣP . This gives us:

Hk(D; A) ∼= Hk(P 00; A) = Hk(Γ0; A)

Hk(C; A) ∼= Hk(E; A) = Hk(G; A)

Hk(Q; A) ∼= Hk(ΣP 0; A) ∼= Hk−1(P 0; A) = Hk(Γ, Γ0; A)

Following Proposition 53 of [20], this will serve as the deﬁnition of the cohomology of a pair (Γ, Γ0). In particular, Proposition 1.2 from [1] gives

0 Proposition 4.1. Let (Γ, Γ ) be a pair as above, C a Γ-projective resolution of Z, D a 0 subcomplex of C which is a Γ-projective resolution of ZΓ/Γ such that D ,→ C induces 0 : ZΓ/Γ ,→ Z and that Q := C/D is Γ-projective. Then the cohomology sequences of C modulo D and of Γ modulo Γ0 are isomorphic. That is, for any Γ-module V , the following diagram commutes up to sign.

··· / Hk(Γ, Γ0; V ) / Hk(Γ; V ) / Hk(Γ0; V ) / Hk+1(Γ, Γ0; V ) / ···

=∼ =∼ =∼ =∼ ··· / Hk(Q; V ) / Hk(C; V ) / Hk(D; V ) / Hk+1(Q; V ) / ··· . 40 Similarly,

k 0 k k 0 k+1 0 ··· / Hb (Γ, Γ ; V ) / Hb (Γ; V ) / Hb (Γ ; V ) / Hb (Γ, Γ ; V ) / ···

=∼ =∼ =∼ =∼ k k k k+1 ··· / Hb (Q; V ) / Hb (C; V ) / Hb (D; V ) / Hb (Q; V ) / ··· . We will apply this later when we have a pair (Γ, Γ0) realized by an Eilenberg-

MacLane pairs (X,Y ), where X is a K(Γ, 1) complex and Y is a subcomplex whose

0 components, Yi, are K(Γi, 1). When this happens, we say (X,Y ) is a K(Γ, Γ ; 1). Bieri and Eckmann formalize this in their paper [1] as Theorem 1.3:

Theorem 4.2. Let Γ be a group and Γ0 a collection of subgroups of Γ. If (X,Y ) is a

K(Γ, Γ0; 1), then

Hk(Γ, Γ0; A) ∼= Hk(X,Y ; A)

Hk(Γ; A) ∼= Hk(X; A)

Hk(Γ0; A) ∼= Hk(Y ; A)

This happens, in particular, when X and Y are aspherical. As mentioned before, the proof also holds for bounded cohomology, however, we no longer need X to be a

K(Γ, 1); it suﬃces for π1(X) = Γ. Similarly for Y . This follows from the fact that the bounded cohomology of a space is isomorphic to the bounded cohomology of any other space with the same fundamental group, an immediate consequence of Lemma

2.31. Thus, we get the similar isomorphisms as above:

Theorem 4.3. Let X be a complex with π1(X) = Γ and Y = tYi a subcomplex

0 whose components Yi have fundamental groups π1(Yi) = Γi. Set Γ = {Γi}. Then the following isomorphisms hold:

k 0 ∼ k ∼ k 0 0 Hb (Γ, Γ ; A) = Hb (X,Y ; A) = Hb (X ,Y ; A)

k ∼ k ∼ k 0 Hb (Γ; A) = Hb (X; A) = Hb (X ; A)

k 0 ∼ k ∼ k 0 Hb (Γ ; A) = Hb (Y ; A) = Hb (Y ; A).

41 CHAPTER 5

CONDITION (B)

We will now return to Lemma 1.3, which we recall here:

Lemma 1.3 Suppose we can ﬁnd a locally CAT(0) n-manifold N which decomposes as N = A ∪f B, where A and B are both truncated, complete, ﬁnite-volume, non-compact, hyperbolic n-manifolds with toric cusps, glued together via f : ∂A → ∂B so that A ∩ B = ∪T n−1. Suppose also that there exists α ∈ H1(A ∩ B) with

∗ 1 1 a) hαi ∩ i (H (A; Z) ⊕ H (B; Z)) = 0

2 2 b) δ(hαi) ∩ im(Hb (N; Z) → H (N; Z)) 6= ∅

We consider this lemma with A = B and look at its the double, N = DA. So far,

42 we have reduced this question to proving the following: Let A be a truncated, complete, ﬁnite volume, non-compact, hyperbolic n-manifold with toric cusps. Consider the following diagram:

2 2 Hb (DA, A) −−−→ Hb (DA)     y y H1(A) −−−→ H2(DA, A) −−−→ H2(DA) −−−→ H2(A)     y y ∗ j∗ H1(A) ⊕ H1(A) −−−→i H1(∂A) −−−→δ H2(DA) −−−→ H2(A) ⊕ H2(A)

Our goal is to show the existence of an element α ∈ H1(∂A) such that

a.) hαi ∩ i∗(H1(A) ⊕ H1(A)) = 0

2 2 b.) δ(hαi) ∩ im(Hb (DA) → H (DA)) 6= ∅

This is equivalent to saying that we want to ﬁnd an element, δ(αk), of inﬁnite order in the intersection of the images of δ and the comparison map on the second cohomology of DA.

· α being in the image of the comparison map is used to tell us that the two

graph manifolds we construct have quasi-isometric fundamental groups.

· α having inﬁnite order tells us that the graph manifold, M2, cannot support a locally CAT(0) metric.

We will see that satisfying condition (a) guarantees that condition (b) is satisﬁed.

To work towards showing that condition b) is satisﬁed, we will show

(i) j(hαi) ⊆ ker(H2(DA) → H2(∂A))

(ii) = im(H2(DA, ∂A) → H2(DA))

(iii) 2 2 = im(Hb (DA) → H (DA)).

43 Containment (i) and equalities (ii) and (iii) will be explained now.

First, the equality (ii) is immediate from the exactness of the long exact sequence for the pair (DA, ∂A).

We now look at the containment (i). We have that α ∈ H1(∂A) and δ(α) ∈

H2(DA), where δ is the coboundary map in the Mayer-Vietoris long exact sequence for the pair (A, A), ie, the two copies of A used to construct DA. δ(α) is then mapped to H2(∂A) by the map in the long exact sequence of the pair (DA, ∂A), induced by inclusion ∂A ,→ DA.

H1(DA) H1(DA)

δ j∗ H1(A) ⊕ H1(A) / H1(∂A) / H2(DA) / H2(A) ⊕ H2(A)

H2(DA, ∂A) / H2(DA) / H2(∂A) /

H3(DA)

The accompanying diagram shows the long exact sequence for the pair (DA, ∂A)

(in dotted arrows) and the Mayer-Vietoris long exact sequence for (A, A) (in dashed arrows). Again, (A, A) are the two copies of A used to construct DA. We have

δ(α) ∈ H2(DA) and we wish to show that j(α) 7→ 0 ∈ H2(∂A) via the long exact sequence for the pair.

From the exactness of the Mayer-Vietoris sequence, we know that j∗(δ(α)) =

0, but by the deﬁnition of the map j∗, this is just the restriction of δ(α) to A, so certainly the further restriction of δ(α) to ∂A is also zero. However, the map

H2(DA) → H2(∂A) is just this restriction, so the image of δ(α)) under this map is 44 0, telling us that δ(α) ∈ ker(H2(DA) → H2(∂A)), thus establishing the containment

(i). We now turn our attention to proving claim (iii), which states:

2 2 2 2 im(H (DA, ∂A) → H (DA)) = im(Hb (DA) → H (DA))

[⊇]

2 2 2 Let x ∈ im(Hb (DA) → H (DA)). There is some y ∈ Hb (DA) that maps to x under the comparison map. Then consider the following diagram

2 2 Hb (DA) / Hb (∂A)

H2(DA, ∂A) / H2(DA) / H2(∂A), where the rows are long exact sequences for the pair (DA, ∂A) for bounded cohomology and ordinary cohomology. The square in the diagram commutes. This is because the horizontal maps are restriction maps, while the vertical maps just forget that a cochain is bounded.

We wish to show that x ∈ im(H2(DA, ∂A) → H2(DA)). From the exactness of the bottom row, it suﬃces to show that x ∈ H2(DA) is in the kernel of the map

H2(DA) → H2(∂A). From the commutativity of the diagram, it suﬃces to show

2 2 2 that y 7→ 0 via the maps Hb (DA) → Hb (∂A) → H (∂A). We can see this to be true using facts about bounded cohomology.

We recall that the bounded cohomology of a group is isomorphic to the bounded cohomology of any space with isomorphic fundamental group (see Lemma 2.31),

n−1 as well as the fact that since Z is amenable, it’s bounded cohomology vanishes (Lemma 2.30). This tells us that

2 Hb (∂A) = 0.

Therefore, we see that y 7→ 0 ∈ H2(∂A), so x ∈ ker(H2(DA) → H2(∂A)) = im(H2(DA, ∂A) → H2(DA)), as desired. 45 2 2 2 2 [⊆] We now show that im(H (DA, ∂A) → H (DA)) ⊆ im(Hb (DA) → H (DA)). To see this, we return to the previous diagram, highlighting a slightly diﬀerent portion of the long exact sequences:

2 2 ··· / Hb (DA, ∂A) / Hb (DA) / ···

··· / H2(DA, ∂A) / H2(DA) / ··· .

Again, the diagram commutes. Because of this commutativity, it suﬃces to show that

2 2 Hb (DA, ∂A) H (DA, ∂A) is surjective. Assume this map is surjective, and let x ∈ im(H2(DA, ∂A) → H2(DA)), so there is some y ∈ H2(DA, ∂A) with y 7→ x. From

2 surjectivity, there exists somey ˆ ∈ Hb (DA, ∂A) such thaty ˆ 7→ y. By commutativity

2 2 of the diagram,y ˆ 7→ xˆ ∈ Hb (DA) andx ˆ 7→ x, showing that x ∈ im(Hb (DA) → H2(DA)), as desired. We now need only to show the map above is surjective:

2 2 Claim 1. The map Hb (DA, ∂A) H (DA, ∂A) is surjective.

In order to prove this claim, we will recall a few results that will help us. The

ﬁrst result, which can be found in its entirety in Theorem 59 in [20], includes the following.

Theorem 5.1 (Mineyev, Yaman). Let Γ be a group and Γ0 be a family of its subgroups.

The following statements are equivalent.

(a) (Γ, Γ0) is relatively hyperbolic pair as in A.6.

0 0 n 0 (e’) There exists a ﬁne tuple (Γ, Γ ,X,V ) of type F and the map Hb (Γ, Γ ; V ) →

n 0 H (Γ, Γ ; V ) is surjective for all bounded QΓ-modules V and all n ≥ 2

46 Proof of claim 1. consider the following diagram

∼ 2 = 2 H (DA, ∂A) / H (π1(DA), π1(∂A)) b (1) b

(4) (2) surj

∼ 2 = 2 H (DA, ∂A) / H (π1(DA), π1(∂A)) (3) It will suﬃce to show

2 2 (1) The map Hb (DA, ∂A) → Hb (π1(DA), π1(∂A)) is an isomorphism.

2 2 (2) The map Hb (π1(DA), π1(∂A)) → H (π1(DA), π1(∂A)) is surjective.

2 2 (3) The map H (DA, ∂A) → H (π1(DA), π1(∂A)) is an isomorphism.

(4) The diagram commutes.

(3) is a direct result of theorem 4.2. (1) results from theorem 4.3. (2) follows from theorem 5.1, using the fact that Γ is hyperbolic relative to Γ0, as was shown in 3.11.

(4) can be seen by looking at the various chain maps. The horizontal isomorphisms are deﬁned in the same way on cochains as they are on bounded cochains, while the vertical maps, which just ignore the fact that cochains are bounded, do not actually change the cochains.

Lemma 5.2. Let A be a locally CAT(0) graph manifold and consider the double DA.

As in lemma 1.3, If we can ﬁnd α ∈ H1(∂A) with property (a),

1 1 hαi ∩ i(H (A; Z) ⊕ H (A; Z)) = 0, then it will automatically satisfy property (b):

2 2 δ(hαi) ∩ im(Hb (DA; Z) → H (DA; Z)) 6= 0.

Hence, from lemma 1.3, we get the answer of yes to problem 1.2.

47 CHAPTER 6

COMPLETING THE PROOF

In their paper [8], Frigerio, Lafont, and Sisto show that condition (a) is equivalent to a homological condition, which allows for more geometric arguments to show the existence of such examples. This is formalized in Lemma 12.3 of [8] as follows.

Lemma 6.1. Let A be a ﬁnite volume hyperbolic manifold such that all boundary components of A are diﬀeomorphic to tori. Then the following two conditions are equivalent:

(1) There exists a non-trivial element α ∈ H1(∂A) which satisﬁes

∗ 1 1 1 hαi ∩ i (H (A; Z) ⊕ H (A; Z)) = {0} ⊂ H (∂A).

(2) i∗ : H1(∂A; Z) → H1(A; Z) is not injective.

Then, they show that this second condition is satisﬁed in Corollary 12.5 of [8]:

Corollary 6.2. (FrlaSi 12.5) The map i∗ : H1(∂A; Z) → H1(A; Z) is not injective.

These result are enough to answer the question posed in 1.2.

Theorem 6.3. There exists inﬁnitely many examples of pairs of graph manifolds with quasi-isometric fundamental groups, with the property that one of them supports a locally CAT(0) metric, but the other one cannot support any locally CAT(0) metric.

Furthermore, these examples arise in every dimension n ≥ 4. 48 Proof. We can now combine lemma 6.1 with corollary 6.2 to conclude that for a finite volume non-compact, hyperbolic manifold, A, with toric cusps, then we can find an element α ∈ H1(∂A) satisfying condition (a). Then, applying lemma 5.2, we can conclude that α also satisfied condition (b). Finally, by lemma 1.3, we are able to conclude the existence of a pair of graph manifolds, with quasi-isometric fundamental groups, such that one of them supports a locally CAT(0) metric, while the other cannot. All the remains is to show that we can find infinitely many such examples. It is sufficient to have the existence of finite volume, non-compact, hyperbolic manifolds with toric cusps in all dimensions, n ≥ 3. This is a result from McReynolds, Reid, and Stover [17].

49 Appendix A

VARIOUS DEFINITIONS OF RELATIVELY

HYPERBOLIC GROUPS

In chapter 3, the definition of a group being hyperbolic relative to a finite collection of subgroups was given. This appendix explores other various equivalent definitions. the following definition was given in chapter 3

0 Deﬁnition A.1. Let Γ be a group and Γ = {Γi|i ∈ I} a family of subgroups of Γ. Γ is hyperbolic relative to Γ0 if there exists a graph K with vertex set V on which Γ acts with the following properties:

• Γ is ﬁnitely generated.

• I is ﬁnite, Γi is ﬁnitely generated for each i.

• K is ﬁne and has thin triangles.

• There are ﬁnitely many orbits of edges and each edge stabilizer is ﬁnite.

0 0 • There exists a Γ-invariant subset V such that V∞ ⊆ V ⊆ V and the stabilizers

0 in V are precisely Γi and their conjugates.

Where V∞ is the set of vertices of inﬁnite valence.

In their paper, Mineyev and Yaman present another definition, which they show to be equivalent to the previous. Before we give this second definition, we first remind ourselves of Mineyev and Yaman’s definition of a tuple. 50 Definition A.2. A graph tuple is a list (Γ, Γ0, G, V 0) such that

• Γ is a group.

0 • Γ = {Γi|i ∈ I} a family of subgroups of Γ, possibly with repetition.

• G is a graph with a Γ-action.

0 0 • V is a Γ-invariant subset such that V∞ ⊆ V ⊆ V .

0 • For each Γi ∈ Γ, there is some vi ∈ V such that stab(vi) = Γi and Γi 7→ Γvi is a bijection between Γ0 and the set of Γ-orbits in V 0.

Finally, a tuple is a list (Γ, Γ0,X,V 0) such that X is a simplicial complex and (Γ, Γ0,X(1),V 0) is a graph tuple.

We now prepare for a second (equivalent) deﬁnition of a hyperbolic pair. The following deﬁnitions, as well as notation, comes from [20].

Definition A.3. (snake metric) Let G be a graph. Two edges e1, e2 ∈ E are called admissible if they have a common vertex a ∈ V. Say e1 = (a, b) and e2 = (a, c). In this case, the angle, anga(e1, e2) is defined to be the length of the shortest path between b and c in G\{a}. If there is no path between b and c in G\{a}, the angle is defined to be ∞.

0 0 Now, let e, e be arbitrary edges and the sequence (e = e0, e1, . . . , en = e ) a sequence of edges such that ei, ei+1 are an admissible pair for i = 0, . . . , n−1. Then the n−1 X angle length of this sequence is ang(ei, ei+1). The snake metric dζ : E × E → Z, i=0 deﬁned on the edge set E of G, assigns to each pair of edges e, e0 the minimal angle

0 length of all admissible sequences (e = e0, e1, . . . , en = e ).

Deﬁnition A.4. A graph G has ﬁne triangles if there exists δ = δ(G) ∈ [0, ∞) with the following property: 51 Given an arbitrary 4(a, b, c) ∈ G with α a geodesic between b and c, β a geodesic between a and c, e1 and edge in α and e2 an edge in β with d(c, e1) = d(c, e2) < 1 (d(a, c) + d(b, c) − d(a, b)), then d (e , e ) ≤ δ. 2 ζ 1 2

0 0 Definition A.5. A tuple (Γ, Γ ,X, V ) is finitely generated, or of type F2, if the Γ action on X(1) is finitely generated in the following sense:

• X(1) is connected.

• X(1) \{v} is connected for each v ∈ V (X(1)).

• There are only ﬁnitely many Γ-orbits in V (X(1)) and in E(X(1)).

• The stabilizers of the edges in E(X(1)) are ﬁnite.

The tuple has fine triangles if X(1) has fine triangles. The tuple is fine is X(1) is fine.

Definition A.6. A tuple (Γ, Γ0,X,V 0) is hyperbolic if it is finitely generated, has fine triangles, and is fine. A pair (Γ, Γ0) is hyperbolic if there exists a hyperbolic tuple.

It is important to note that this deﬁnition of relative hyperbolicity agrees with

Deﬁnition 3.1. This is given in Theorem 39 from [20]:

Theorem A.7. The following statements are equivalent.

(a) (Γ, Γ0) is a hyperbolic pair in the sense of Deﬁnition A.6.

(b) Γ is hyperbolic relative to Γ0 in the sense of Deﬁnition 3.1.

The following result is due to Mineyev and Yaman [20], relating the notion of relative hyperbolicity and the comparison map from bounded group cohomology to standard group cohomology. While the full statement can be found in [20], we include only the portion necessary for our purposes.

52 Theorem A.8. Let Γ be a group and Γ0 be a family of its subgroups. Then the following are equivalent:

(a) (Γ, Γ) is hyperbolic as in deﬁnition A.6.

0 0 n 0 (e’) There exists a ﬁne tuple (Γ, Γ ,X,V ) of type F and the map Hb (Γ, Γ ; V ) →

n 0 H (Γ, Γ ; V ) is surjective for all bounded QΓ-modules V and all n ≥ 2.

We now shift to a diﬀerent point of view of relative hyperbolicity, due to Farb.

In his paper [7], he defines a coned off Cayley graph as well as the bounded coset penetration (BCP) property. These two properties combine to give an alternate, equivalent definition of relative hyperbolicity.

Definition A.9. Let G be a group and H a finitely generated subgroup of G. Let Γ be the Cayley graph of G. The coned off Cayley graph of G with respect to H, denoted ˆ ˆ Γ = Γ(H), is formed as follows. Add a vertex, vgH for each left coset gH of H in G.

Then add an edge of length 1/2 from gh to vgH for each h ∈ H.

The deﬁnition generalizes in the natural way for multiple subgroups. Farb gives some remarks on the coned oﬀ Cayley graph.

• Γˆ is not a proper metric space.

• In general, Γˆ is very diﬀerent from the quotient graph obtained by quotienting

out by the left action of H on Γ, due to the diﬀerence between left and right

cosets.

q.i. • If H/G, then Γ(ˆ H) ∼ Γ(H\G).

• Γˆ is quasi-isometric to the graph obtained by identifying each left coset to a

point in Γ.

53 Deﬁnition A.10. The group G is called weakly hyperbolic relative to {H1,H2,...,Hn} if the coned-oﬀ Cayley graph Γˆ is a hyperbolic metric space.

Following [4], we note that Farb’s deﬁnition in [7] says that if this is the case, then G is hyperbolic relative to H. However, Γˆ being negatively curved is a weaker condition than Bowditch’s deﬁnition. In fact, in [25], Szczepanski proves that if a group G is hyperbolic relative to {H1,...,Hn} in the sense of Gromov, then G is weakly hyperbolic relative to {H1,...,Hn} ([25], Thm 1). Additionally, Szczepanski gives an example to show that these two classes of groups are in fact not the same

([25], Example 3). We continue with [23] to see what information is missing from the deﬁnition of weak relative hyperbolicity.

Definition A.11. Let G be a finitely generated group, X a finite generating set. ˆ Given a path u in Γ, u penetrates the coset gHi if u passes through the vertex corresponding to the coset gHi, called the cone point. The vertex of the path u before the cone point is called the entering vertex, while the vertex of the path u after the cone point is called the exiting vertex. The entering and exiting vertices will always be vertices of Γ. The path u is said to be without backtracking if it penetrates each cone point at most once.

Definition A.12. Let G be a finitely generated group, X a finite generating set, Λ the Cayley graph of G with respect to X. The pair (G, {H1,...,Hn}) satisfies the bounded coset penetration (BCP) property if, for every λ ≥ 1, there is a constant c = c(λ) > 0 such that if u, v are λ-quasi-geodesics without backtracking in Λˆ such that the endpoints of u and v are in Λ, u− = v− and d(u+, v+) ≤ 1, then the following conditions hold

• If u penetrates a coset gHi but v does not penetrate gHi, then the entering vertex

54 and the ending vertex of u in gHi are an X-distance of at most c from each other.

• If both u and v penetrate a coset gHi, then the entering vertices of u and v

in gHi are at X-distant at most c from each other. Similarly for the exiting vertices.

Deﬁnition A.13. Let G be a ﬁnitely generated group and {H1,...,Hn} a collection of subgroups of G. G is hyperbolic relative to {H1,...,Hn} in the sense of Farb if

G is weakly hyperbolic relative to {H1,...,Hn} and (G, {H1,...,Hn}) has the BCP property.

Dahmani [5] shows that Gromov’s and Farb’s deﬁnitions of relative hyperbolicity agree. A proof can also be found in [4] and [23].

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