Dimension of Virtually Cyclic Classifying Spaces for Certain Geometric Groups

DIMENSION OF VIRTUALLY CYCLIC CLASSIFYING SPACES FOR CERTAIN GEOMETRIC GROUPS

DISSERTATION

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

in the Graduate School of the Ohio State University

Kyle Joecken, M.S.

Graduate Program in Mathematics

The Ohio State University

2013

Dissertation Committee:

Jean-Fran¸coisLafont, Advisor

Nathan Broaddus

Mike Davis

Ivonne Ortiz c Copyright by

Kyle Joecken

2013 ABSTRACT

Given a connected, oriented, closed 3-manifold M, we construct models for EΓ, the classifying space of Γ = π1(M) with isotropy in the virtually cyclic subgroups; we also compute the smallest possible geometric dimension for EΓ, pointing out in which cases the models are larger than necessary.

This is done by decomposing M using the prime and JSJ decompositions; the resulting manifolds are either closed and geometric or compact with geometric interior by Thurston’s

Geometrization Conjecture. We develop a pushout construction of models for virtually cyclic classifying spaces of fundamental groups of Seifert ﬁber spaces with base orbifold modeled on H2, then—using a pushout method of Lafont and Ortiz ([LO09b])—we combine these with known models for the remaining pieces to obtain a model for EΓ. These models are then analyzed using Bredon cohomology theory to see if they are of the smallest possible dimension.

ii To Serena: Thanks for waiting.

iii ACKNOWLEDGMENTS

I must ﬁrst thank my advisor Jean-Fran¸coisLafont for his patience, encouragement, energy, and time over the last three years. Without a doubt, his willingness to go the extra mile with me is the largest reason I am in a position to submit this work.

I would also like to thank Fangyang Zheng and Cindy Bernlohr for conversations that were extremely helpful along the way. Their guidance was crtical to my existence in the program.

I would like to thank Ivonne Ortiz and Dan Farley, who with my advisor gave me the opportunity to give a talk on preliminary results at the 2012 AMS Fall Central Sectional

Meeting. That experience helped solidify my work, and resulted in both a useful professional experience and signiﬁcant feedback.

My oﬃcemate Tim All has been a constant and willing sounding board; our discussions about our work—though unrelated—helped foster a creative and productive environment.

Letting me voice my thoughts out loud to someone was useful to my distilling process. Thank you, Tim.

Finally, I would like to thank my friends and family for their patience and understanding.

They gave me love, encouragement, and space when needed. In particular I’d like to thank my wife Serena for her extreme patience and understanding in letting me ﬁnish this goal.

iv VITA

1982 ...... Born

2004 ...... B.S. in Mathematics, Case Western Reserve University

2006 ...... M.S. in Mathematics The Ohio State University

2004-Present ...... Graduate Teaching Associate, The Ohio State University

FIELDS OF STUDY

Major Field: Mathematics

Specialization: Algebraic Topology

v TABLE OF CONTENTS

Abstract ...... ii

Dedication ...... iii

Acknowledgments ...... iv

Vita...... v

CHAPTER PAGE

1 Preliminaries ...... 6

1.1 Models for Classifying Spaces ...... 6 1.2 CAT(κ) Spaces ...... 9 1.3 Isometries ...... 12 1.4 Geometric and Cohomological Dimension ...... 18 1.5 Classifying Spaces for Crystallographic Groups ...... 34 1.6 Amalgams and Bass-Serre Trees ...... 36

2 Virtually Cyclic Model Construction Methods ...... 44

2.1 Virtually Cyclic Extensions ...... 44 2.2 Acylindrical Splittings ...... 51

3 Virtually Cyclic Models for Geometric Groups ...... 54

3.1 Seifert Fiber Spaces ...... 54 3.1.1 Compact base orbifolds with nonempty boundary ...... 59 3.1.2 Closed base orbifolds modeled on H2 ...... 64 3.1.3 Closed base orbifolds modeled on E2 ...... 71 3.2 Compact Manifolds modeled on Sol or H3 ...... 77 4 Compact 3-Manifold Group Classiﬁcation ...... 89

4.1 Further Questions ...... 103

Bibliography ...... 108

vi INTRODUCTION

The goal of this paper is to compute the smallest possible geometric dimension of a model for EΓ = EVCΓ, where Γ is the fundamental group of a closed oriented 3-manifold; we denote this quantity gd(Γ). We do this by ﬁrst studying gd(Γ) when Γ is the fundamental group

3 2 3 of a compact 3-manifold modeled on one of the eight Thurston geometries: S , S × E, E , 3 2 H , PSLg 2(R), H × E, Nil, and Sol. It is known that any closed oriented 3-manifold can be expressed as the connected sum of a unique set of prime oriented 3-manifolds; that is, manifolds that cannot themselves be nontrivially expressed as a connected sum. Thurston’s

Geometrization Conjecture states that each prime manifold can be cut along a (possibly trivial) collection of pairwise disjoint embedded tori so that the interiors of each of the resulting pieces is modeled on a Thurston geometry or is Seifert ﬁbered; in 2002-2003, Grigori

Perelman proved the ﬁnal piece of this conjecture. It is thus the goal of this paper to ﬁrst demonstrate how to construct models for EΓ in the cases without a known construction.

Then, using the pushout construction in [LO09b], we combine models for the pieces into models ﬁrst for the manifolds of the prime decomposition, then into a model for EΓ. Finally, with knowledge of the minimal dimension of the individual pieces, we attempt to compute the minimal dimension for a model for EΓ.

The most prevalent place in which models for EΓ appear is the Farrell-Jones Isomorphism

Conjecture. Given a group Γ, one can deﬁne a generalized equivariant homology theory by

1 redeﬁning the homology of the single point Γ-space {∗} (that is, by changing the dimen-

sion axiom in the Eilenberg-Steenrod axioms). In particular, one can deﬁne a generalized

equivariant homology theory by setting

Γ −∞ Hn ({∗}; KR ) = Kn(RΓ)

for a given group ring RΓ. One could then use this generalized equivariant homology to

at best compute the algebraic K-theory of RΓ, or at least use knowledge of generalized

homology theory to create relationships between the K-theory of group rings of related

groups (e.g., amalgamations, extensions).

Suppose one has a Γ-space X; then the map X → {∗} is trivially Γ-equivariant. This

map induces what is known as an assembly map homomorphism:

Γ −∞ Γ −∞ Hn (X; KR ) −→ Hn ({∗}; KR ) = Kn(RΓ).

The Farrell-Jones Isomorphism Conjecture ([FJ93]) suggests that if X is a model for EΓ, then this map is actually an isomorphism. The conjecture is in general still open, but has been shown to hold for a wide range of classes of groups (see [LR05] for several examples); in particular, it has been shown for 3-manifold groups ([Rou08, BFL13]).

In Chapter 1 we discuss the necessary preliminaries.

In Chapter 2 we adapt pushout constructions of Lafont and Ortiz given in [LO09a,

LO09b], both of which are based on a construction given in [LW12]. First, let F ⊆ F 0 be

families of subgroups of a group Γ, and let φ :Γ → Γ0 be a surjective homomorphism onto

a quotient Γ0 of Γ. We demonstrate that, with knowledge of a model for Eφ(F)Γ0 and a few

additional models, one can construct a model for EF 0 Γ. We are particularly interested in the families FIN and VC of ﬁnite and virtually cyclic subgroups of Γ, respectively; given

a discrete group Γ and a surjective homomorphism Γ → Γ0 with virtually cyclic kernel,

2 we show that for a collection of subgroups He ∈ H of Γ that are φ-preimages of particular

subgroups H ≤ Γ0, the cellular Γ-pushout X constructed via:

a Γ × EH EΓ He 0 He∈H

a Γ × EH X He e He∈H

is a model for EΓ.

Similarly, we demonstrate that if a graph of groups (Γ,Y ) exists with respect to which

Γ is acylindrical, then then Γ-CW -space X contructed via the cellular Γ-pushout

a G ×H E EF G H∈H

a G ×H {∗} X H∈H

is a model for EΓ; here the family F consists of subgroups of Γ that are conjugate to a

subgroup of one of the vertex subgroups in (Γ,Y ).

In Chapter 3 we use the above methods to construct models for EΓ, where Γ is the

fundamental group of a compact Seifert ﬁber space. In the case where M is a Seifert ﬁber

2 space modeled on PSLg 2(R) or H × E, these models are 3-dimensional; this is smaller than previously constructed models (e.g. [Far10]). We then show that gd(Γ) = 3; that is, that

our models are of minimal dimension. For completeness, we also demonstrate that if Γ ≤

Isom(X) is the fundamental group of a connected, oriented, compact manifold modeled on

the geometry X, then

3 2 • gd(Γ) = 0 if X is S or S × E (as Γ is virtually cyclic);

3 • gd(Γ) = 4 if X is E (cf. [CFH08]); 3 • gd(Γ) = 3 otherwise.

Finally, in Chapter 4 we attempt to construct models for EΓ when Γ is the fundamental group of a connected, closed, oriented 3-manifold M. We do this by attempting to piece together models for the geometric and Seifert ﬁbered manifolds that appear in the decomposition of M conjectured by Thurston.

In particular, the Seifert-van Kampen Theorem tells us that a decomposition of a manifold M along disjoint incompressible surfaces gives rise to a group of graphs (Γ,Y ) whose fundamental group Γ is isomorphic to π1(M). We demonstrate that for the prime decomposition and JSJ decomposition, the corresponding graph of groups gives a splitting of Γ that is acylindrical. Letting F be the family of subgroups of Γ that conjugate into one of the vertex groups of (Γ,Y ), we construct a model for EF Γ from virtually cyclic models for the vertex groups, each of which is the fundamental group of a geometric or Seifert ﬁbered manifold that is compact. We then apply the second contruction of Chapter 2 to obtain a model for EΓ and use it to bound gd(Γ) as much as possible:

Theorem. Let M be a connected, closed, oriented 3-manifold. Let M = P1# ... #Pk be the

1 2 prime decomposition of M into closed, irreducible manifolds and copies of S × S . For each closed, irreducible manifold Pi, let Ti be a minimal collection of incompressible tori whose removal decomposes Pi into connected compact Seifert ﬁbered or hyperbolic manifolds. Let

Γ = π1(M) be the fundamental group of M. Then gd(Γ) ≤ 4; we can classify gd(Γ) by the first set of conditions satisfied in the list below (in particular, M satisfies at least one of the following):

3 1. If any of the irreducible Pi are modeled on E , then gd(Γ) = 4;

2. If any of the irreducible Pi have a JSJ decomposition along a (nonempty, disjoint)

collection Ti of essential tori, then 3 ≤ gd(Γ) ≤ 4;

4 3 2 3. If any of the irreducible Pi are modeled on one of H , H × E, PSLg 2(R), Sol, or Nil, then gd(Γ) = 3;

4. If k > 2 or if M = P1#P2 with |π1(P1)| > 2, then gd(Γ) = 2;

5. If Γ is virtually cyclic, then gd(Γ) = 0.

5 CHAPTER 1

PRELIMINARIES

1.1 Models for Classifying Spaces

We begin by establishing the terminology surrounding the objects we wish to construct.

Deﬁnition. Let Γ be a discrete group. A nonempty set F of subgroups of Γ is called a

family if, given A ∈ F and γ ∈ Γ, we have that B ≤ γ−1Aγ ⇒ B ∈ F (that is, F is closed

under conjugation and passing to subgroups).

Remark 1.1. It is clear from this deﬁnition that if φ :Γ → Γ0 is a group homomorphism

and F is a family of subgroups of Γ0, then Fe := {A ≤ Γ | φ(A) ∈ F} is a family of subgroups of Γ.

Given a group Γ, useful families are TR (the family consisting of only the trivial sub-

group), ALL (the family of all subgroups of Γ), and FIN (the family of ﬁnite subgroups).

We are particularly interested in VC, the family of virtually cyclic subgroups (i.e., subgroups having a cyclic subgroup of ﬁnite index). In particular, we have a nice fact about virtually cyclic groups:

Lemma 1.2. Let G be an inﬁnite virtually cyclic group; then there exists a normal subgroup ∼ of ﬁnite index K E G with K = Z.

6 Proof. Because G is inﬁnite virtually cyclic, we know there is an inﬁnite cyclic subgroup

H ≤ G of finite index, say H = hhi for some h ∈ G. We define K to be the core of H in G: \ K = g−1Hg. g∈G It is clear by definition that K E G, so it only remains to show that [G : K] < ∞ and that ∼ K = Z.

Since [G : H] < ∞ and H ≤ NG(H) ≤ G, we have that there are only ﬁnitely many conjugates of H in G. As such, we can rewrite k \ −1 K = gi Hgi i=1

for some g1, . . . , gk ∈ G; we may as well take g1 = 1. Since each conjugate of H is also of

−1 finite index in G, we have that H ∩gi Hgi has finite index in H. Since there are only finitely ∼ many conjugates, their intersection K must also be of finite index in H = Z, and therefore

must be isomorphic to Z itself. Finally, since [G : K] = [G : H] · [H : K], K is ﬁnite index in G.

Deﬁnition. Let Γ be a group. A Γ-CW-complex is a CW-complex with a Γ-action that

sends cells to cells, and such that an element of Γ leaves a cell invariant only if it ﬁxes it

pointwise.

Deﬁnition. Let Γ be a group and F a family of subgroups of Γ; we call a Γ-CW-complex a model for EF Γ if for every H ≤ Γ:

1. H/∈ F ⇒ XH = ∅ (XH is the H-ﬁxed subcomplex of X);

2. H ∈ F ⇒ XH is contractible.

The simplest example is when F = ALL consists of all subgroups of Γ, where a point is

a model for EALLΓ. Models for ETRΓ are contractible spaces with a free Γ-action; they are often denoted simply EΓ.

7 Models for EFIN Γ (which we abbreviate to EΓ) are known for many classes of discrete groups Γ. We adapt known methods [LO09a, Far10, LW12] to construct models for EVCΓ (which we abbreviate to EΓ).

We have a couple simple but useful results about transference of models.

Proposition 1.3. Let φ :Γ → Γ0 be a homomorphism between discrete groups and let F be a family of subgroups of Γ0. Let E be a model for EF Γ0. Then E with Γ-action induced by φ is a model for E Γ as well. Fe

Proof. Let A ∈ Fe; then φ(A) ∈ F, so EA = Eφ(A), which is contractible. If A is not in Fe then φ(A) cannot be in F, since A ≤ φ−1(φ(A)); thus EA = Eφ(A) = ∅ and E is a model for

E Γ, as claimed. Fe

Corollary 1.4. Let Γ be a discrete group, H a subgroup of Γ. Let F be a family of subgroups

of Γ and let F|H be the family {F ∈ F | F ≤ H} of subgroups of H. Let E be a model for

EF Γ; then E is a model for EF|H H.

Proof. Let φ : H → Γ be the embedding of H into Γ; then Fe = F|H . Apply Proposition 1.3.

Remark 1.5. If Γ is a subgroup, F is a family of subgroups of Γ, and H is a subgroup of

Γ, we will from here on denote the classifying space EF|H H as simply EF H.

One ﬁnal object of use in the building of models is the Borel construction.

Deﬁnition. Let Γ be a discrete group, H ≤ Γ, F a family of subgroups of H, and EF H a model for the classifying space of H with isotropy in the family F. Then the Borel

construction Γ ×H EF H is the Γ-CW -complex (with Γ-action given by left multiplication)

that is the quotient of the direct product Γ × EF H by the diagonal H-action (where the H-action on Γ is given by right multiplication).

8 Remark 1.6. A geometrically useful way of thinking of Γ ×H EF H is as a disjoint union

of copies of EF H—one for each left coset of H in Γ—along with a Γ-action that is given in

the following way; let {γi}i∈I be a set of left coset representatives of H in Γ. Then for each

−1 representative γi, there is a representative γj such that γγi ∈ γjH, or γj γγi ∈ H. We let

−1 γ act on Γ ×H EF H by ﬁrst acting via γj γγi ∈ H on the copy associated to each γi, then

permuting the copies by taking the copy associated to each γi to the copy associated to γj.

1.2 CAT(κ) Spaces

For a more complete study, see Bridson and Haeﬂiger ([BH99, Chapters I.2, II.1, II.8]).

n Deﬁnition. Given a real number κ, we denote by Mκ the following metric spaces:

n n n • if κ = 0 then M0 is Euclidean space E (i.e., R with the Euclidean metric);

n n • if κ > 0 then Mκ is obtained from the standard unit sphere S by multiplying the √ distance function by the constant 1/ κ;

n n • if κ < 0 then Mκ is obtained from hyperbolic space H by multiplying the distance √ function by 1/ −κ.

n Mκ is the n-dimensional model space of constant curvature κ. Deﬁne the constant Dκ to be √ Dκ := π/ κ if κ > 0 and Dκ := ∞ otherwise; Dκ is called the diameter of the model space

n Mκ .

Deﬁnition. Let X be a metric space, and let κ be a positive real number. We call X a geodesic space if all points x, y ∈ X are joined by a geodesic line. We say X is a Dκ-geodesic space if all points x, y ∈ X that satisfy d(x, y) < Dκ are joined by a geodesic.

n The Mκ are called model spaces as they give us a way to deﬁne curvature for ‘suﬃciently’ geodesic metric spaces.

9 Lemma 1.7 ([BH99, Lemma I.2.14]). Let κ be a real number, and let p, q, r be three points in a metric space X satisfying d(p, q) + d(q, r) + d(r, p) < 2Dκ. Then there exist points

2 p,¯ q,¯ r¯ ∈ Mκ such that d(p, q) = d(¯p, q¯), d(q, r) = d(¯q, r¯) and d(r, p) = d(¯r, p¯).

2 The triangle ∆(¯p, q,¯ r¯) ⊆ Mκ with vertices p,¯ q,¯ r¯ is called a comparison triangle for the

2 triple (p, q, r); it is unique up to isometries of Mκ . If ∆ ⊆ X is a geodesic triangle with vertices p, q, r, then ∆(¯p, q,¯ r¯) is also said to be a comparison triangle for ∆. Given any

2 x ∈ X on the geodesic from p to q in ∆, a comparison pointx ¯ ∈ Mκ is a point along the geodesic from p¯ to q¯ in ∆¯ = ∆(¯p, q,¯ r¯) satsfying d(p, x) = d(¯p, x¯) (and thus necessarily d(x, q) = d(¯x, q¯)).

Deﬁnition. Let X be a metric space and let κ be a real number. Let ∆ be a geodesic ¯ 2 triangle in X with perimeter less than 2Dκ. Let ∆ ⊂ Mκ be a comparison triangle for ∆. Then ∆ is said to satisfy the CAT(κ) inequality if for all x, y ∈ ∆ and all comparison points x,¯ y¯ ∈ ∆,¯ d(x, y) ≤ d(¯x, y¯).

If κ ≤ 0, X is called a CAT(κ) space if X is a geodesic space all of whose geodesic triangles satisfy the CAT(κ) inequality. If κ > 0, then X is called a CAT(κ) space if X is

Dκ-geodesic and all geodesic triangles in X of perimeter less than 2Dκ satisfy the CAT(κ) inequality. (Here we allow for the possibility that the metric on X may take inﬁnite values.)

Remark 1.8. Note that for κ ≤ κ0, a CAT(κ) space is automatically a CAT(κ0) space.

Moreover, by multiplying the distance function by an appropriate factor, any CAT(κ) space can be thought of as a CAT( − 1), CAT(0), or CAT(1) space for κ < 0, κ = 0 or κ > 0, respectively.

Deﬁnition. A metric space X is said to be of curvature ≤ κ if it is locally a CAT(κ) space; i.e., for every x ∈ X there exists an rx > 0 such that the ball B(x, rx)—endowed with the induced metric—is a CAT(κ) space.

10 There are a number of useful results on CAT(κ) spaces that come directly from the deﬁnitions ([BH99, Proposition II.1.4]).

Proposition 1.9. Let X be a CAT(κ) space.

1. There is a unique geodesic segment joining each pair of points x, y ∈ X satisfying

d(x, y) < Dκ, and this geodesic segment varies continuously with its endpoints.

2. Every local geodesic in X of length at most Dκ is a geodesic.

3. The balls in X of radius less than Dκ/2 are convex; i.e., any two points in such a ball are joined by a geodesic segment contained in the ball.

4. The balls in X of radius less than Dκ are contractible.

For CAT(0) spaces, the above can be reﬁned.

Corollary 1.10. Let X be a CAT(0) space.

1. X is uniquely geodesic; i.e., there is a unique geodesic line joining any two points

x, y ∈ X, and the geodesic varies continuously with x and y.

2. Every local geodesic is a geodesic.

3. Balls in X of ﬁnite radius are convex.

4. X is contractible; in particular it is simply connected and all higher homotopy groups

are trivial.

We now wish to discuss the boundary at inﬁnity of a CAT(0) space.

Deﬁnition. Let X be a metric space. Two geodesic rays c, c0 : [0, ∞) → X (or two

0 geodesic lines c, c : R → X) are asymptotic if there exists some constant K ∈ R such that d(c(t), c0(t)) ≤ K for all t. We deﬁne the set ∂X of boundary points of X to be the set

of equivalence classes of geodesic rays (two rays being equivalent if they are asymptotic). The

equivalence class of a geodesic ray c is denoted c(∞). We denote by X¯ the union X ∪ ∂X. 11 Bridson and Haeﬂiger demonstrate existence and uniqueness of rays in a complete CAT(0)

space ([BH99, Proposition II.8.2]).

Proposition 1.11. If X is a complete CAT(0) space and c : [0, ∞) → X is a geodesic ray

issuing from x, then for every point x0 ∈ X there is a unique geodesic ray c0 which issues

from x0 and is asymptotic to c.

A stronger result holds if X is a complete CAT(−1) space (see [BH99, Remark II.9.12(1),

Proposition II.9.21(1)]).

Proposition 1.12. Let X be a complete CAT( − 1) space, ξ, ξ0 ∈ ∂X distinct. Then there

0 exists a geodesic c : R → X with c(∞) = ξ, c(−∞) = ξ .

Remark 1.13. If X is a complete CAT(0) space, there exist two metrics on ∂X, and ∂X

0 is a CAT(1) space with respect to either of them. The angular metric ∠(ξ, ξ ) between two boundary points ξ, ξ0 ∈ ∂X is the supremum of the angle between any two geodesic rays c, c0

0 0 0 in X satisfying c(0) = c (0), c(∞) = ξ, c (∞) = ξ . The Tits metric (denoted dT ) is the length metric associated to the angular metric. See [BH99, Chapter II.9] for more details.

n n Of particular interest to us is the boundary ∂H ; given any point x ∈ H , note that the n n−1 n sphere {y ∈ H | d(x, y) = 1}' S is in natural bijection with the boundary ∂H by identifying the end c(∞) of each geodesic ray c issuing from x with the point c(1). Using

this same idea with the unit tangent sphere around any point in a nonpositively curved

n−1 Riemannian n-manifold X, one can see that ∂X ' S .

1.3 Isometries

Many of the discrete groups Γ we consider are realized as subgroups of isometry groups

acting on metric spaces. The deﬁnitions and results in this section again come mainly from

Bridson and Haefliger ([BH99, Chapter II.6]). 12 Definition. Let X be a metric space and let γ be an isometry of X. The displacement function of γ is the function dγ : X → R≥0 defined by dγ(x) = d(γ.x, x). The translation length of γ is the number |γ| := inf{dγ(x): x ∈ X}. The set of points where dγ attains this infimum is denoted Min(γ). More generally, if Γ is a group acting by isometries on X, then \ Min(Γ) := Min(γ). γ∈Γ An isometry γ is called semi-simple if Min(γ) 6= ∅. An action of a group of isometries of

X is called semi-simple if all its elements are semi-simple.

The displacement function has some nice properties.

Proposition 1.14. Let X be a metric space and let γ be an isometry of X. Let Γ be a group acting by isometries on X.

1. Min(γ) is γ-invariant and Min(Γ) is Γ-invariant.

2. If α is an isometry of X, then |γ| = |αγα−1| and Min(αγα−1) = α.Min(γ).

3. If X is a CAT(0) space, then the displacement function dγ is convex, and Min(Γ) is therefore a closed convex set.

Deﬁnition. Let X be a metric space, Γ ≤ Homeo(X). We say that the action of Γ is:

• eﬀective (or alternatively faithful) if the only element of Γ acting trivially is the identity

element;

• free if no point of X is ﬁxed by an element of Γ other than the identity;

• discrete if Γ is a discrete subset of Isom(X) with the compact-open topology;

• properly discontinuous if X is locally compact, and for every compact subset K of X

the set of γ ∈ Γ such that γK ∩ K 6= ∅ is ﬁnite.

• cocompact if Γ acts properly discontinuously in such a way that the quotient space

X/Γ is compact. 13 • coﬁnite if X is a Riemannian manifold and Γ acts properly discontinuously in such a

way that the quotient space X/Γ has ﬁnite volume.

If Γ acts on X discretely and coﬁnitely, we call Γ a lattice in Isom(X); if Γ additionally acts cocompactly, we call Γ a uniform lattice. Often the group Γ itself is referred to as coﬁnite or cocompact if its action on X has the corresponding property.

The following result ([BH99, Proposition II.6.10(2)]) tells us what types of isometries are found in a uniform lattice.

Proposition 1.15. Suppose the group Γ acts properly cocompactly by isometries on a metric space X. Then Γ acts semi-simply on X.

Remark 1.16. Isometries that have a ﬁxed point are called elliptic; isometries that attain a nonzero min d(x, γ.x) are called hyperbolic. Isometries that are not semi-simple are called x∈X parabolic.

We will generally restrict our work to CAT(0) spaces. Let us ﬁrst consider elliptic isometries.

Deﬁnition. A metric space X is proper if for every x ∈ X and r > 0, the closed ball

B¯(x, r) = {y ∈ X | d(x, y) ≤ r} is compact.

Lemma 1.17. Let X be a proper CAT(0) space, and let Γ ≤ Isom(X) act properly discontinuously on X. Then Γx (the stabilizer in Γ of x) is ﬁnite for all x ∈ X.

Proof. Let x ∈ X. By deﬁnition of a properly discontinuous action on a locally compact space, for any compact K ⊆ X,

|{γ ∈ Γ | γ(K) ∩ K 6= ∅}| < ∞.

Apply this to K = {x}.

14 Additionally, ﬁnite (hence elliptic) groups of isometries must have a shared ﬁxed point

([BH99, Corollary II.2.8(1)]):

Proposition 1.18. If X is a complete CAT(0) space and Γ is a ﬁnite group of isometries of X, then the ﬁxed-point set of Γ is a nonempty convex subspace of X.

Corollary 1.19. Let X be a complete CAT(0) space, H a ﬁnite group of isometries of X; then XH is contractible.

H H H Proof. By Proposition 1.18 X is nonempty and convex; let x0 ∈ X . Then for any y ∈ X we have (by Corollary 1.10) a unique geodesic cy(R) ⊂ X with cy(0) = y, cy(1) = x0; since H H H H X is convex, cy(t) ∈ X for t ∈ [0, 1]. Then f : X ×[0, 1] → X deﬁned by f(y, t) = cy(t)

H H is a homotopy satisﬁng f(X , 0) = idXH , f(X , 1) ≡ x0 (note that f is continuous due to the convexity of the distance function). Thus, XH is contractible.

Hyperbolic isometries have been characterized by Bridson and Haeﬂiger ([BH99, Theorem

II.6.8]).

Theorem 1.20. Let X be a CAT(0) space.

1. An isometry γ of X is hyperbolic if and only if there exists a geodesic line c : R → X which is translated nontrivially by γ; i.e., γ.c(t) = c(t + a) for some a > 0. The set

c(R) is called an axis of γ. For any such axis, the number a is actually equal to |γ|.

2. If X is complete and γm is hyperbolic for some integer m 6= 0, then γ is hyperbolic.

Let γ be a hyperbolic isometry of X.

3. The axes of γ are parallel to each other and their union is Min(γ).

4. Min(γ) is isometric to Y ×R, and the restriction of γ to Min(γ) is of the form (y, t) 7→

(y, t + |γ|), where y ∈ Y, t ∈ R.

15 5. Every isometry α that commutes with γ leaves Min(γ) = Y × R invariant, and its 0 00 0 00 restriction to Y × R is of the form (α , α ), where α is an isometry of Y and α is a

translation of R.

As isometries ﬁx distances, they behave nicely on asymptotic equivalence classes of geodesic rays, and therefore on ∂X ([BH99, Corollary II.8.9]).

Proposition 1.21. Let γ be an isometry of a complete CAT(0) space X. The natural extension of γ to X¯ is a homeomorphism of X¯.

We wish to understand inﬁnite virtually cyclic subgroups of isometries acting discretely and properly discontinuously on a proper CAT(0) space. To do so we ﬁrst give a theorem known as the Flat Strip Theorem ([BH99, Theorem II.2.13]).

0 Theorem 1.22. Let X be a CAT(0) space, and let c : R → X and c : R → X be geodesic 0 0 lines in X. If c and c are asymptotic, then the convex hull of c(R) ∪ c (R) is isometric to a 2 ﬂat strip R × [0,D] ∈ E .

Corollary 1.23. Let X be a CAT( − 1) space, and let ξ, ξ0 ∈ ∂X be distinct boundary points

0 of X. Then the geodesic c(R) ⊂ X satisfying c(∞) = ξ and c(−∞) = ξ is unique (up to parametrization).

Proof. Existence is given by Proposition 1.12. If it weren’t unique, X would contain a ﬂat strip given by Theorem 1.22. Then the CAT(−1) inequality fails for any nondegenerate triangle in the strip, a contradiction.

Corollary 1.24. Let X be a proper CAT(0) space, γ a hyperbolic isometry of X stabilizing

0 two geodesic lines c(R) and c (R). Then γ acts by translation on the convex hull of c(R) ∪ 0 c (R). In particular if X is a proper CAT( − 1) space then γ stabilizes a unique geodesic in X.

16 Proof. For CAT(0) spaces, the claim follows from Theorem 1.20. The convex hull is ﬂat

according to Theorem 1.22; nondegenerate triangles in a ﬂat strip violate the CAT(−1)

inequality, so such geodesics must be unique in CAT(−1) spaces.

Infinite virtually cyclic subgroups behave even more nicely on CAT(−1) spaces. We want to work with both uniform and nonuniform lattices, so we’ll need to allow for parabolic isometries; Bridson and Haefliger offer the following result ([BH99, Proposition II.8.25]).

Proposition 1.25. Let γ be an isometry of a proper CAT(0) space X. If γ is parabolic, then it ﬁxes at least one point ξ ∈ ∂X.

Corollary 1.26. A parabolic isometry γ of a proper CAT( − 1) space X ﬁxes a unique boundary point ξ ∈ ∂X.

Proof. By Proposition 1.25, γ ﬁxes at least one boundary point ξ ∈ ∂X. Suppose γ also

ﬁxes some other point ζ ∈ ∂X; then by Corollary 1.23 there is a unique geodesic c(R) with c(∞) = ξ and c(−∞) = ζ. But then γ must stabilize c(R), contradicting the assumption that γ is parabolic; we conclude that γ ﬁxes only ξ in ∂X.

Proposition 1.27. Let X be a proper CAT( − 1) space, H ≤ Isom(X) an inﬁnite virtually cyclic subgroup acting properly discontinuously on X. Then either H stabilizes a unique geodesic c(R) ⊂ X or ﬁxes a unique point ξ ∈ ∂X.

Proof. Since H is inﬁnite and virtually cyclic, there exists some h ∈ H of inﬁnite order satisfying [H : hhi] < ∞. By Lemma 1.17, h cannot be elliptic.

Suppose h is hyperbolic; then by Corollary 1.24, h stabilizes a unique geodesic c(R) ⊂ X. Let γ ∈ H; then hγhγ−1i is an inﬁnite subgroup of H. Since hhi is of ﬁnite index in H,

−1 n n −1 k n −1 there exist nonzero h, k ∈ Z such that (γhγ ) = γh γ = h . Thus, γh γ is a

hyperbolic isometry stabilizing both c(R) and γ.c(R). Again by Corollary 1.24 we have that

γ.c(R) = c(R), so that γ stabilizes c(R). As γ was arbitrary, we have shown that H stabilizes

c(R). 17 Now suppose h is parabolic; then by Corollary 1.26 h ﬁxes a unique point ξ ∈ ∂X. By

the same conjugating argument a generic γ ∈ H must also ﬁx ξ, so H ﬁxes ξ as a group.

The full converse is not true, but the following partial converse will prove useful.

Proposition 1.28. Let X be a proper CAT( − 1) space, Γ ≤ Isom(X) a lattice. Let H ≤ Γ be an inﬁnite subgroup for which all elements stabilize some geodesic line c(R) ⊂ X. Then H is virtually cyclic.

Proof. Note that H acts semi-simply on X. Let Hc be the set of elements of H that pointwise

fix c; then Hc EH. Moreover, since Hc is contained in the group Γc(0)—isometries in Γ fixing c(0) ∈ X—which is finite by Lemma 1.17, |Hc| < ∞. Let Q = H/Hc and let f : H → Q be the canonical homomorphism. Since Hc pointwise fixes c, there is an induced effective ∼ isometric action of Q on E; that is, Q ≤ Isom(R) = R o Z2. Moreover, Q acts discretely and cocompactly on R, so Q is congruent to either Z or D∞. Let h ∈ H satisfy |f(h)| = ∞; then |h| = ∞ and

[H : hhi] = [Q : hf(h)i] · |Hc| < ∞.

Thus H is virtually cyclic.

1.4 Geometric and Cohomological Dimension

We wish to show some of the models we construct are of smallest possible dimension. To

that end, we have the following deﬁnition.

Deﬁnition. Let G be a discrete group and let F be a family of subgroups of G. The geometric dimension of G for the family F—denoted gdF (G)—is the smallest dimension of a model for EF G. If no ﬁnite-dimensional model exists, we say that gdF (G) = ∞.

Remark 1.29. As with models for classifying spaces, we leave oﬀ the trivial family (gdTR = gd), we use a single underline to denote that the associated family is the family of ﬁnite 18 subgroups (gd), and we use a double underline for the family of virtually cyclic subgroups

(gd).

Constructing an n-dimensional model for E G forces gd (G) ≤ n, and knowing gd (H) F F F|H for a subgroup H ≤ G gives a lower bound gd (H) ≤ gd (G) by Corollary 1.4. In the case F|H F where G is the fundamental group of a closed 3-manifold, these two facts combined will often be suﬃcient to compute gdF (G) explicitly. However, there are times when no subgroup is easily found with gd as large as that of the constructed model.

Fortunately there is another measurement of dimension that is closely related to gdF , known as the Bredon cohomological dimension. To define it, we first need to define Bredon cohomology, which requires some tools from category theory. Most of the basics of category theory are culled from [Jac80b], while the results related to Bredon cohomology come from the treatment in [MV03, Part I Section 3].

Deﬁnition. A category C is an algebraic structure that starts with a collection of objects

(denoted Ob(C)). Between any pair M,N of objects (not necessarily distinct) we have a collection of morphisms (denoted Mor(M,N)) that satisfy two properties:

• (Closure of composition) For any M,N,P ∈ Ob(C), if σ ∈ Mor(M,N) and τ ∈

Mor(N,P ) then τ ◦ σ ∈ Mor(M,P );

• (Existence of identities) For any M ∈ Ob(C), idM ∈ Mor(M,M) composes trivially with any morphism into or out of M; that is, if M,N ∈ Ob(C) with f ∈ Mor(M,N),

then f ◦ idM = f = idN ◦ f.

Given objects M,N,P ∈ Ob(C), a morphism ϕ ∈ Mor(M,N) is an epimorphism if for any

σ, τ ∈ Mor(N,P ) with σ ◦ ϕ = τ ◦ ϕ, we have σ = τ; this is a generalization of surjective functions. A morphism ϕ ∈ Mor(N,P ) is a monomorphism if for any σ, τ ∈ Mor(M,N) with ϕ◦σ = ϕ◦τ, we have σ = τ; this is a generalization of injective functions. A morphism

19 ϕ ∈ Mor(M,N) is an isomorphism if there exists a morphism ϕ−1 ∈ Mor(N,M) (called an

−1 −1 −1 inverse) satisfying ϕ ◦ ϕ = idM and ϕ ◦ ϕ = idN ; clearly ϕ is also an isomorphism.

Remark 1.30. From this point forward, to indicate X is an object in C we will say X ∈ C rather than X ∈ Ob(C).

Deﬁnition. Let C be a category, and suppose that for any pair of objects A, B ∈ C that the set Hom(A, B) has the structure of an abelian group. Then C is called preadditive.

As an example let R be any ring, and deﬁne the category R to have the single object R with Hom(R,R) the set of endomorphisms (homomorphisms from R to itself). By giving this set the binary operation (σ + τ)(r) = σ(r) + τ(r), we see that Hom(R,R) has the structure of an abelian group, so that R is a preadditive category.

Deﬁnition. Given a group G and family of subgroups F, the orbit category OF G has as objects the spaces of left cosets G/H with H ∈ F. Given a pair G/H and G/K of left coset spaces, we deﬁne the morphisms ϕ ∈ Mor(G/H, G/K) to be those functions that satisfy

ϕ(gH) = gϕ(H) for any g ∈ G.

We can characterize the morphisms in OF G by their image of H ∈ G/K; if ϕ(H) = xK for some x ∈ G, then for all g ∈ G, ϕ(gH) = gxK; we denote this morphism ϕx. To be

−1 well-deﬁned, if h1H = h2H (that is, if h2 h1 ∈ H), we should have

h1xK = ϕx(h1H) = ϕx(h2H) = h2xK.

−1 H This implies h2 h1xK = xK, or that xK ∈ (G/K) (the natural action of H on G/K ﬁxes xK).

H On the other hand, for each left coset xK in (G/K) we have a morphism ϕx ∈ Mor(G/H, G/K); the choice of representative x is clearly irrelevant. So the sets (G/K)H and Mor(G/H, G/K) are in bijection, and are usually taken to be the same.

20 Deﬁnition. Given two categories C and D, a (covariant) functor F is a map from C to D that takes objects to objects and morphisms to morphisms in a way that satisﬁes:

• If ϕ ∈ Mor(X,Y ) is a map between objects in C, then F (ϕ) ∈ Mor(F (X),F (Y )) with

F (X),F (Y ) ∈ D;

• If σ ∈ Mor(X,Y ) and τ ∈ Mor(Y,Z) for some X,Y,Z ∈ C then F (τ ◦σ) = F (τ)◦F (σ)

in Mor(F (X),F (Z)).

A contravariant functor G from C to D is the same as above, except that it reverses morphisms. Thus, the two conditions it must satisfy on morphisms become:

• If ϕ ∈ Mor(X,Y ) is a map between objects in C, then F (ϕ) ∈ Mor(F (Y ),F (X)) with

F (X),F (Y ) ∈ D;

• If σ ∈ Mor(X,Y ) and τ ∈ Mor(Y,Z) for some X,Y,Z ∈ C then F (τ ◦σ) = F (σ)◦F (τ)

in Mor(F (Z),F (X)).

Finally, if F1 and F2 are two functors from C to D, a natural transformation Φ: F1 → F2

associates to each object M ∈ C a morphism ΦM ∈ Mor(F1(M),F2(M)) between objects in D (called the component of Φ at M), so that if ϕ ∈ Mor(M,N) for any M,N ∈ C, the

following diagram commutes:

F1(ϕ) F1(M) F1(N)

ΦM ΦN

F2(ϕ) F2(M) F2(N).

If G1 and G2 are contravariant functors from C to D, we simply reverse the horizontal arrows

in the commutative diagram Φ : G1 → G2:

G1(ϕ) G1(M) G1(N)

ΦM ΦN

G2(ϕ) G2(M) G2(N).

21 We can now deﬁne a category where the objects are functors and the morphisms are the

associated natural transformations.

Deﬁnition. Let C be a preadditive category. A right C-module M (resp. left) is a covariant

(resp. contravariant) functor from C to the category Z-Mod of abelian groups. The categories C-Mod of right C-modules and Mod-C of left C-modules have as morphisms the natural

transformations.

Remark 1.31. If R is the category given by a ring R, then these are the usual left and right

R-modules. Note that Z-modules are in one-to-one correspondence with abelian groups; for this reason you often see orbit modules as functors into A, the category of abelian groups. For

a given Z-module A, we denote by A ∈ Mod-OF G the constant functor satisfying A(G/H) =

A for all H ∈ F and A(φ) = idA for all φ ∈ Mor(G/H, G/K).

Let G be a group. We deﬁne a right OF G-module M (resp. left) as a covariant (resp. con-

travariant) functor from OF G to Z-Mod. The categories OF G-Mod of right OF G-modules

and Mod-OF G of left OF G-modules have as morphisms the natural transformations. Though

OF G is not preadditive (in general some sets of homomorphisms are empty), these categories behave much the same, as we will see.

We would like to construct exact sequences in a category of modules; in order to do this,

we need the category to be abelian.

Deﬁnition. Let C be a category. An object I ∈ C is initial if for any M ∈ C, there is a

unique ϕ ∈ Mor(I,M). An object T ∈ C is terminal if for any M ∈ C, there is a unique

ϕ ∈ Mor(M,T ). A zero object is both initial and terminal (think of the trivial group in the category of abelian groups).

Given M,N ∈ C, the (binary) product of M and N is the object M × N ∈ C along with

a pair of morphisms πM ∈ Mor(M × N,M) and πN ∈ Mor(M × N,N), such that if P is

22 another object with morphisms fM ∈ Mor(P,M) and fN ∈ Mor(P,N) then there exists a unique morphism f ∈ Mor(P,M × N) such that the following diagram commutes:

fM fN f M M × N N. πM πN

The (binary) coproduct of M and N is the object M ⊕N ∈ C along with a pair of morphisms

ιM ∈ Mor(M,M ⊕ N) and ιN ∈ Mor(N,M ⊕ N), such that if C, gM , gN is another such grouping then there exists a unique morphism g ∈ Mor(M ⊕ N,C) such that the following diagram commutes:

C f f M g N

M M ⊕ N N. ιM ιN

In a preadditive category, products and coproducts coincide; we call an object that is both a product and a coproduct a biproduct. A preadditive category is called additive if it contains all ﬁnite biproducts (equivalently, if it contains a zero object (null biproduct) and all binary biproducts).

Deﬁnition. Let C be a category. Let M,N ∈ C, ϕ ∈ Mor(M,N). The morphism ϕ is a zero morphism from M to N (denoted 0MN ) if ϕ satsﬁes both of the following conditions:

• for any K ∈ C and σ, τ ∈ Mor(K,M), σ ◦ ϕ = τ ◦ ϕ (ϕ is a left zero morphism);

• for any P ∈ C and σ, τ ∈ Mor(N,P ), ϕ ◦ σ = ϕ ◦ τ (ϕ is a right zero morphism).

Suppose now that C has zero morphisms. Let M,N ∈ C and ϕ ∈ Mod(M,N). The kernel of ϕ is an object K ∈ C and monomorphism k ∈ Mor(K,M) such that ϕ ◦ k = 0KN ,

23 and if K0 ∈ C, k0 ∈ Mor(K0,M) is another such object then there exists a unique morphism

u ∈ Mor(K0,K) such that the following diagram commutes:

M ϕ k 0 k K 0KN N u

0K0N K0

Conversely, the cokernel of ϕ is an object Q ∈ C and an epimorphism q ∈ Mor(N,Q) such

0 0 0 that q◦ϕ = 0MQ, and if Q , q are another such pair then there exists a unique u ∈ Mor(Q, Q ) such that the following diagram commutes:

N ϕ q 0 0MQ q M Q u

0MQ0 Q0

An additive category that contains all kernels and cokernels is preabelian.

A monomorphism ϕ is normal if it is the kernel of some morphism; an epimorphism is called conormal if it is the cokernel of some morphism. A category is normal (resp. conormal) if all its monomorphisms (resp. epimorphisms) are normal (resp. conormal); a category that is both normal and conormal is called binormal.

A preabelian category that is also binormal is called abelian; in other words, an abelian category is a preadditive category with a zero object, zero morphisms, all ﬁnite biproducts, all kernels and cokernels, and in which every monomorphism is a kernel of some morphism and every epimorphism is the cokernel of some morphism.

It is clear from basic ring theory that R-Mod and Mod-R are abelian categories.

Claim 1.32. For a discrete group G, Mod-OF G and OF G-Mod are abelian categories. 24 Proof. The module 0 that takes all coset spaces to the trivial Z-module is a zero object in both categories. Trivial homomorphisms always exist and are zero morphisms.

Let Φ ∈ Mor(M,N) be a morphism between M,N ∈ Mod-OF G; then for all H ∈ F,

ΦH ∈ Mor(M(G/H),N(G/H)) is a group homorphism. Deﬁne K(G/H) := ker ΦH , and let kH : K(G/H) → M(G/H) be the injection. Then kH is one-to-one, and so is a monomor-

phism; moreover, kH ◦ ΦH is the zero map. The universal property is similarly clear, as

0 0 0 any other group K (G/H) and map kH with kH ◦ ΦH the zero map has an obvious map

0 0 u : K (G/H) → K(G/H) via u(g) = kH (g) ∈ K(G/H). For any ϕ ∈ HomF (G/H → G/L), deﬁne K(ϕ): K(G/L) → K(G/H) so that the following diagram commutes:

K(G/H) kH M(G/H) ΦH N(G/H)

K(ϕ) M(ϕ) N(ϕ)

K(G/L) kL M(G/L) ΦL N(G/H).

This can be done by considering that each x ∈ K(G/L) satisﬁes N(ϕ) ◦ Φ(L) ◦ KL(x) =

0 ∈ N(G/H), so M(ϕ) ◦ kL(x) ∈ ker ΦL. Thus, choosing y ∈ K(G/H) to be the unique

element with kH (y) = M(ϕ) ◦ kL(x), we can set K(ϕ)(x) = y. Letting K ∈ Mod-OF G and k ∈ Mor(K,M) be thusly deﬁned, we see that ϕ has a kernel. The argument is virtually

the same for M,N ∈ OF G-Mod; only the roles of H and L are reversed in the derivation of K(ϕ). Cokernels may be similarly deﬁned by components.

Given two modules M,N ∈ OF G, we deﬁne their product and coproduct componentwise;

M × N(G/H) = M(G/H) × N(G/H) with morphisms (πM )H = πM(G/H) and (πN )H =

πN(G/H) and similarly for M ⊕ N. The necessary properties of the associated morphisms come immediately componentwise as they did for kernels.

Finally, let Φ ∈ Mor(K,M) be a monomorphism for K,M ∈ Mod-OF G. Then ΦH : K(G/H) → M(G/H) is injective for every H ∈ F, and since the groups are abelian,

K(G/H) E M(G/H) by this embedding. Thus, we may deﬁne a module N ∈ Mod-OF G by N(G/H) = M(G/H)/K(G/H) and a morphism f : M → N which is the canonical

25 quotient map on each component; this map is made to be a natural transformation by choosing appropriate homomorphisms N(ϕ): G/H → G/L for each ϕ ∈ HomF (G/H, G/L) in an argument similar to the one for the existance of kernels. Φ is then the kernel of f; the same construction works for right OF G-modules, and a similar one works to prove that epimorphisms Φ : M → N are cokernels by deﬁning K(G/H) := ker ΦH , kH the embedding on each component.

Deﬁnition. Let C be a preabelian category; then a sequence

ϕ1 ϕ2 ϕn−1 M1 → M2 → ... → Mn

is called exact if ker ϕi = cokerϕi+1 for 1 ≤ i ≤ n − 2. If C is abelian, then an exact sequence of the form

f g 0 → A → B → C → 0 is called a short exact sequence. Note that by the deﬁnition of zero objects, f is a monomorphism and g is an epimorphism.

For A, B, C ∈ Mod-OF G, f ∈ Mor(A, B) and g ∈ Mor(B,C), to say the sequence

f g 0 → A → B → C → 0 is exact is to say the sequence

f g 0 → A(G/H) →H B(G/H) →H C(G/H) → 0 is exact in Z-Mod for all H ∈ F.

Deﬁnition. Let C, D be abelian categories, let F : C → D be a covariant functor and let

G : C → D be a contravariant functor. Let 0 → A →ι B →π C be exact. Then:

F (ι) F (π) • F is a left exact functor if the sequence 0 → F (A) → F (B) → F (C) is exact;

26 G(π) G(ι) • G is a left exact functor if the sequence G(C) → G(B) → G(A) → 0 is exact.

Similarly, if A →ι B →π C → 0 is exact, we say:

F (ι) F (π) • F is a right exact functor if the sequence F (A) → F (B) → F (C) → 0 is exact;

G(π) G(ι) • G is a right exact functor if the sequence 0 → G(C) → G(B) → G(A) is exact.

If F (or G) is left and right exact—that is, if it takes short exact sequences to short exact sequences—then we say that the functor is exact.

Let M ∈ Mod-OF G; we deﬁne the functor

HomF (M, −) : Mod-OF G → Z-Mod

via N 7→ HomF (M,N), the abelian group of all natural transformations from M to N. Similarly, we deﬁne the contravariant functor

HomF (−,M) : Mod-OF G → Z-Mod

via N 7→ HomF (N,M).

Claim 1.33. Let M ∈ Mod-OF G. Then HomF (−,M) is right exact and HomF (M, −) is left exact.

f g Proof. The proof is an exercise is diagram chasing. Let A → B → C → 0 be an exact sequence of modules in Mod-OF G; then for every H ∈ F, the sequence

f g A(G/H) →H B(G/H) →H C(G/H) → 0 of Z-modules is exact. The contravariant functor HomF (−,M) gives rise to a sequence

g¯ f¯ 0 → HomF (C,M) → HomF (B,M) → HomF A(G/H).

We need to show thatg ¯ is injective, that ker f¯ ⊆ img ¯, and that img ¯ ⊆ ker f¯.

27 Suppose that σ, τ ∈ HomF (C,M) are such thatg ¯(σ) =g ¯(τ) (see diagram below). Let

H ∈ F, and let c ∈ C(G/H). Then gH is surjective given the short exact sequence in which

it lies, so there is some b ∈ B(G/H) with gH (b) = c. Sinceg ¯(σ)H =g ¯(τ)H and the above diagram commutes, we have that

σH (c) = σH (gH (b)) =g ¯(σ)H (b) =g ¯(τ)H (b) = τH (gH (b)) = τH (c).

So σH = τH for all H ∈ F, andg ¯ is therefore injective.

B(G/H)

g¯(σ)H g¯(τ)H gH

M(G/H) C(G/H) M(G/H) σH τH

¯ Next, suppose σ ∈ HomF (B,M) is such that f(σ) ≡ 0. We want to construct a τ ∈

HomF (C,M) such thatg ¯(τ) = σ (see diagram below). Let H ∈ F and let c ∈ C(G/H);

then there exists a b ∈ B(G/H) with gH (b) = c. We want to deﬁne τH (c) = σH (b), but

0 0 need to check that this would be well-deﬁned; let b ∈ B(G/H) satisfy gH (b ) = c. Then

0 f g b − b ∈ ker gH , so since the sequence A → B → C is exact, there is some a ∈ A(G/H) with 0 0 ¯ 0 fH (a) = b − b . But then σH (b − b ) = f(σ)H (a) = 0, so σH (b) = σH (b ) and our choice for

τH (c) is well-deﬁned.

f g A(G/H) H B(G/H) H C(G/H)

σH ¯ τH f(σ)H =0H M(G/H)

Having deﬁned τ on each component, we need to show that it is a natural transformation.

Let H,K ∈ F, φ ∈ Mor(G/H, G/K), and c ∈ C(G/K) (see diagram below). Since gK is

surjective, there is some b ∈ B(G/K) with gK (b) = c. As g and σ are natural transformations, we have that

τH ◦ C(φ)(c) = τH ◦ C(φ) ◦ gK (b) = τH ◦ gH ◦ B(φ)(b) = σH ◦ B(φ)(b)

= M(φ) ◦ σK (b) = M(φ) ◦ τK ◦ gK (b) = M(φ) ◦ τK (c). 28 So τ is a natural transformation satisfyingg ¯(τ) = σ, and thus ker f¯ ⊆ img ¯.

C(φ) C(G/H) C(G/K)

gK g H τK

τH B(G/H) B(G/K) B(φ)

σH σK M(G/H) M(G/K) M(φ) ¯ Finally, we want to show that f ◦ g¯ ≡ 0. Let σ ∈ HomF (C,M), let H ∈ F, and let f g a ∈ A(G/H) (see diagram below). g ◦ f ≡ 0 since A → B → C is exact, so

¯ f ◦ g¯(σ)H (a) =g ¯(σ)H ◦ fH (a) = σH ◦ gH ◦ fH (a) = 0.

f g A(G/H) H B(G/H) H C(G/H)

g¯(σ)H ¯ σH f◦g¯(σ)H M(G/H)

¯ This concludes the proof that HomF (−,M) is right exact. f is not in general surjective, as given a σ ∈ HomF (A, M) there is no a priori way to deﬁne τ ∈ HomF (B,M) outside of

im fH ⊂ B(G/H) on each component.

That HomF (M, −) is left exact is largely a similar diagram chase, starting instead with f g an exact sequence 0 → A → B → C.

Remark 1.34. Given a ring R and left R-module M, we may similarly deﬁne functors

HomR(M, −) and HomR(−,M) that are left and right exact; the argument is analogous to

the above argument for left OF G-modules.

Deﬁnition. A left OF G-module P is projective if the functor HomF (P, −) is exact; equiv-

alently, for any M,N ∈ Mod-OF G and maps φ ∈ Mor(M,N) and σ ∈ Mor(P,N) with φ surjective, there exists a map τ ∈ Mor(P,M) such that τ ◦ φ = σ.

Given a left OF G-module M, an inﬁnite exact sequence

... → Pn → ... → P1 → P0 → M → 0 29 with projective OF G-modules Pi is called a projective resolution admitted by M, and is

denoted P∗(M). The length of P∗(M) is the largest n ∈ Z for which Pn is not the zero module 0; if no such largest nonzero module exists, we say that the resolution has inﬁnite

length.

The deﬁnition above is not restricted to the category OF G; given any ring R, one can deﬁne projective R-modules and projective resolutions admitted by a chosen R-module in the

same way. We say a category has enough projectives if, given any left module M there exists

a projective module P and an epimorphism P → M. One can construct such projectives in

Mod-R and Mod-OF G. In particular note that, given a model X for EF G, there is a natural

projective resolution that comes from the chain complex Ci(X) = Z[∆i] (the free abelian

module generated by ∆i), where ∆i is the set of i-cells in the G-CW -complex X. The set

∆i has a natural G-action induced by the G-action on X; the cochain complex obtained

by applying the (natural extension of the) HomF (−,M) functor to the chain complex is a projective resolution and can be used to compute the Bredon cohomology. In particular, if

X is n-dimensional for some n ∈ Z nonnegative, then ∆i = ∅ for all i > n, and therefore i HF (G; M) = 0 for all i > n and all M ∈ Mod-OF G. Thus, cdF (G) ≤ n, and more generally, cdF (G) ≤ gdF (G). For more details on the construction of this projective resolution, see [MV03, Lemma 3.4].

Returning to Bredon cohomology, the idea is to start with the constant left OF G-module

Z, create a projective resolution

∂2 ∂1 ...P2 → P1 → P0 → Z → 0,

apply the contravariant functor HomF (−,M) for a chosen left OF G-module M (referred to as the coeﬃcients), then compute the (reduced) cohomology groups of the resulting cochain

complex. It is surprising that the choice of projective resolution does not matter; the resulting

Z-modules are dependent only on G, F, and the choice of M. Denoting by P∗(M) a projec- i tive resolution admitted by M, one deﬁnes functors Ext (Z,M): OF G → Z-Mod, known as 30 the right derived functors of the HomF (−,M) functor, as taking the group G to the i-th co-

i i homology group of the above cochain complex; that is, Ext (Z,M) = H (HomF (P∗(Z),M)).

Deﬁnition. Let G be a discrete group, F a family of subgroups of G, M a left OF G-module.

i The Bredon cohomology groups of G with coeﬃcients in M (denoted HF (G; M) for i ≥ 0) are deﬁned to be the right derived functors of the HomF (−,M) functor; that is,

i i HF (G; M) = Ext (Z,M).

Remark 1.35. In the case where F is just the trivial family TR = {{1}}, Bredon coho-

mology collapses to an equivarient cohomology; OTRG = OG consists of only the trivial orbit space G/{1} ∼= G, orbit modules are vaccuously constant, and the Bredon cohomology

i i i HTR(G; M) = H (G; M) is just H (HomG(C∗(X),M(G))) where X is a model for EG (a

contractible free G-CW -complex), C∗(X) is the usual chain complex on X, and M(G) is

some abelian group. If M(G) = Z (i.e., if M = Z), Bredon cohomology over TR reduces to ordinary cohomology Hi(X/G). For further details, see [MV03].

We can now describe a dimension computed by the Bredon cohomology of a group.

Deﬁnition. Let G be a discrete group, F a family of subgroups of G. The Bredon coho-

mological dimension of G—denoted cdF (G)—is the largest nonnegative n ∈ Z for which the n Bredon cohomology group HF (G; M) is nontrivial for some M ∈ Mod-OF G. If no such

largest integer exists, we say that cdF (G) = ∞.

We can deﬁne similar notions of dimension for ordinary cohomology.

Deﬁnition. Let G be a discrete group, and let ZG be the integer group ring. The length of

the shortest projective resolution P∗(Z) of projective ZG-modules is called the cohomological

dimension of G, and is denoted cd(G). If Z admits no ﬁnite-length projective resolution of

ZG-modules, we say that cd(G) = ∞.

31 We deﬁne the virtual cohomological dimension of G to be the inﬁmum of the set

{cd(H) | H ≤ G, [G : H] < ∞};

That is, if G has a ﬁnite index subgroup with ﬁnite cohomological dimension, then vcd(G)

is the smallest value of cd(H) taken over ﬁnite index subgroups H ≤ G; if G has no such

subgroup, vcd(G) = ∞.

A great amount is known about cd(G). We consult [Bro94, Chapter VIII] for some useful

results.

Proposition 1.36 (Corollary 2.5). If cd(G) < ∞, then G is torsion-free.

The following theorem is due to Serre.

Theorem 1.37 (Theorem 3.1). If G is a torsion-free group and G0 is a subgroup of ﬁnite

index, then cd(G) = cd(G0).

Corollary 1.38. Let G be a virtually torsion-free group; then any ﬁnite-index torsion-free

subgroup G0 satisﬁes cd(G) = vcd(G).

Proof. We follow an argument given in [Bro94, VIII.11]. Let G be a virtually torsion-free

group, and let G0 be a torsion-free subgroup of ﬁnite index with cd(G0) = vcd(G). Let G00

be any other ﬁnite-index torsion-free subgroup of G. Then H = G0 ∩ G00 is ﬁnite-index in G0

and G00, so by Theorem 1.37 we have that cd(G0) = cd(H) = cd(G00); the result follows.

For most groups, it turns out that cd(G) = gd(G).

Proposition 1.39. Let G be a group. If cd(G) 6= 2 then cd(G) = gd(G); if cd(G) = 2, then

2 ≤ gd(G) ≤ 3.

Proof. This follows from a theorem of Eilenberg and Ganea which constructs a connected

CW -complex Y of dimension max{cd(G), 3} for which π1(Y ) = G and the universal cover 32 X = Ye is contractible. But then X is a model for EG of the same dimension; in particular, gd(G) ≤ max{cd(G), 3}. Given a G-space X on which G acts freely, we obtain a projective resolution of length n of projective ZG-modules over Z via the chain complex C∗(X); thus, cd(G) ≤ gd(G). That cd(G) = gd(G) if cd(G) = 0, 1 is known, trivially in the ﬁrst case and as a result of a deep theorem by Stallings and Swan in the second; see [Bro94, Corollary 7.2] for more details.

Though vcd(G) is not as well-known, we have at least the following result.

Proposition 1.40. Let G be a group with vcd(G) < ∞; then vcd(G) ≤ gd(G).

Proof. Let G be a group with vcd(G) < ∞; then G has a subgroup G0 with cd(G0) = vcd(G) < ∞; by Proposition 1.36, G0 must be torsion-free. By Theorem 11.1 and Adden- dum 11.2 in [Bro94], since vcd(G) < ∞ there exists a ﬁnite-dimensional model for EG; in particular, gd(G) < ∞. Let X be a model for EG of smallest dimension; then the G-action on X induces a G0-action on X. Moreover, since G0 is torsion-free, this action is free. So

0 we construct a projective resolution of ZG -modules over Z of length gd(G), and therefore vcd(G) = cd(G0) ≤ gd(G), as claimed.

With the following result of L¨uck and Meintrup ([LM00, 0.1a]), we can make a comment about how related cdF (G) and gdF (G) are.

Theorem 1.41. Let G be a discrete group and let d ≥ 3. Then there is a d-dimensional G-

CW-model for EF G if and only if the constant OF G-module Z has a d-dimensional projective resolution.

Again, this leads to the following result:

Corollary 1.42. For any group G and family of subgroups F,

cdF (G) ≤ gdF (G) ≤ max{3, cdF (G)}.

33 i Remark 1.43. As usual, OF (G), HF (G; M), and cdF (G) are all rendered with underlines when F is the family of ﬁnite or virtually cyclic subgroups (e.g., OG, Hi(G; M), and cd(G) when the family consists of the virtually cyclic subgroups).

1.5 Classifying Spaces for Crystallographic Groups

This section discusses the results of a recent paper by Connolly, Fehrman, and Hartglass

([CFH08]).

n Deﬁnition. An n-crystallographic group is a uniform lattice in Isom(E ), n ∈ N.

Such groups were classiﬁed by Bieberbach ([Bie11, Bie12]).

Proposition 1.44. Let Γ be an n-crystallographic group. Then Γ contains a maximal free

n abelian normal subgroup of rank n corresponding to the translations of E in Γ; this subgroup is of ﬁnite index in Γ.

The converse was later shown by Zassenhaus ([Zas48]).

Theorem 1.45. An abstract group Γ is isomorphic to an n-dimensional crystallographic group if and only if G contains a ﬁnite index, normal, free abelian subgroup of rank n, that is also maximal abelian.

∼ n Let Γ be an n-crystallographic group. Then Γ = A o G, where A = Z is the subgroup of translations and G = Γ/A is a discrete subgroup of O(n) (called a crystallographic point group). We wish to construct models for EΓ and EΓ, as well as compute vcd(Γ), gd(Γ), and gd(Γ); all of these pieces will be of use to us in later constructions.

Proposition 1.46. Let Γ be an n-crystallographic group, n ∈ N. Then vcd(Γ) = n.

n n Proof. Γ contains a copy of Z as a ﬁnite-index torsion-free subgroup; cd(Z ) = n. The result follows by Corollary 1.38.

34 n n Claim 1.47. E with Z -action given by linearly independent translations is a model for n EZ of smallest dimension.

n Proof. The only finite subgroup of Z is the trivial subgroup, which fixes the whole (con- n n tractible) space E . Any nontrivial subgroup of Z contains an infinite-order element that n n acts freely on E , so that the fixed subspace of any nontrivial subgroup of Z is empty. n n n n That the above model exists gives gd(Z ) ≤ n; that n = cd(Z ) ≤ gd(Z ) forces gd(Z ) = n (see Proposition 1.40).

∼ n Proposition 1.48. Let Γ be an n-crystallographic group, Γ = Z o G, G ≤ O(n) ﬁnite. n Then E with Γ-action given by translations on the ﬁrst component and the action induced by O(n) about the origin on the second is a model for EΓ of smallest dimension.

n F Proof. A finite subgroup F of Γ has contractible (E ) by Corollary 1.19. An infinite subgroup must contain an element acting by translation, so the fixed subspace is empty.

n By Corollary 1.4 and Claim 1.47, n = gd(Z ) ≤ gd(Γ), so gd(Γ) = n.

We now construct models for EΓ as done in [CFH08]. Note that if Γ is 1-crystallographic,

then Γ is virtually cyclic. EΓ = EALLΓ is thus modeled by a point, and gd(Γ) = 0. We hereafter assume n ≥ 2. ∼ n Let A = Z be the subgroup of translations in Γ. Deﬁne

C := {H ≤ A | H is a maximal cyclic subgroup of A}.

For each subgroup C ∈ C, deﬁne

n−1 n E (C) := {` ∈ E | ` a line,C · ` = `}.

n n−1 n Deﬁne the quotient map πC : E → E (C) to take the point x ∈ E to the unique line n−1 n ` ∈ E (C) containing x. Deﬁne an equivalence relation ∼ on the product E × I × C by letting (x, t, C) ∼ (x0, t0,C0) if

35 1.0 < t = t0 < 1, C = C0 and x = x0;

0 0 0 2. t = t = 1, C = C and πC (x) = πC (x );

3. t = t0 = 0 and x = x0.

n Let E = (E × I × C)/ ∼, and deﬁne a Γ-action on E by

γ · [x, t, C] = [γ · x, t, γCγ−1] for all γ ∈ Γ.

Note that C is a discrete set, so E is (n + 1)-dimensional. Connolly, Fehrman and Hartglass

give the following two results ([CFH08, Proposition 3.1, Theorem 1.1]):

Proposition 1.49. The Γ-space E is a model for EΓ.

Theorem 1.50. If Γ is n-crystallographic with n ≥ 2, gd(Γ) = n + 1.

In particular, the model constructed above is of smallest dimension.

1.6 Amalgams and Bass-Serre Trees

In order to discuss models with virtually cyclic isotropy for a general connected, closed,

oriented 3-manifold, we will need to understand how to build a model for a manifold knowing

models for the pieces of its decomposition. In particular, we’ll need to understand their

fundamental groups.

The following discussion comes largely from [Ser03].

Deﬁnition. Let (Gi)i∈I be a collection of groups, and for each ordered pair (i, j) let Fij be

a set of homomorphisms from Gi to Gj. Deﬁne the direct limit of the Gi relative to the Fij

(denoted G = lim Gi) to be a group G and a set of homomorphisms fi : Gi → G such that −→

fj ◦ f = fi for all f ∈ Fij, with G and homomorphisms fi universal; that is, if there exists

another group H and homomorphisms hi : Gi → H satisfying hj ◦ f = hi for all f ∈ Fij, then there is a unique homomorphism h : G → H such that h ◦ fi = hi for all i ∈ I. 36 We have from [Ser03, Proposition 1] the following:

Proposition 1.51. The pair consisting of G and the family (fi)i∈I exists and is unique up to unique homomorphism.

Consider the relatively simple situation with groups A, G1, and G2 and given monomorphisms f1 : A → G1 and f2 : A → G2; the direct limit G is called the amalgamation of A in

G1 and G2 and is denoted G1 ∗A G2. If A = {1} is the trivial group this corresponds to the free product of G1 and G2, denoted G1 ∗ G2. These deﬁnitions are of use in the statement of the Van Kampen Theorem:

Theorem 1.52. Let X be a topological space covered by two open sets, U1 and U2. Let

V = U1 ∩ U2, and suppose that U1, U2, and V are all pathwise connected. Choose a basepoint

x ∈ V . Then the fundamental group G = π1(X; x) is realized as the amalgamation of

A = π1(V ; x) in G1 = π1(U1; x) and G2 = π1(U2; x) relative to the natural embeddings f1 : π1(V ; x) → π1(U1; x) and f2 : π1(V ; x) → π1(U2; x).

Next we give a definition of a graph slightly different from the usual graph theoretical definition.

Deﬁnition. A graph Y consists of a set X = vert Y , a set E = edge Y , and two maps

E → X × X : y 7→ (o(y), t(y)) and E → E : y 7→ y¯ satisfying y¯ = y,y ¯ 6= y, and o(y) = t(¯y).

An element P ∈ X is called a vertex of E; an element y ∈ E is called an (oriented) edge

of Y , withy ¯ its inverse edge. The vertex o(y) is called the origin of y, and the vertex t(y)

is called the terminus of y; together the vertices o(y) and t(y) are called the extremities of

the edge y. Two vertices are said to be adjacent in Γ if they are the extremities of an edge

y ∈ E.

Given a graph Y 0 with vertices X0 and edges E0, a morphism from Y to Y 0 is a pair

of maps, one mapping vertices to vertices and the other mapping edges to edges, that send

inverses to inverses, origins to origins and termini to termini. 37 a ¯ An orientation of a graph Y is a subset E+ of E such that E = E+ E+; in other words, it is a choice of one of each pair of edges {y, y¯}.

We want to think of these graphs in the usual sense; for this we can consider the realization of the graph.

Deﬁnition. Let Y be a graph with X = vert Y and E = edge Y . We let the topological space T be the disjoint union of X and E × [0, 1], with the discrete topology on X and E.

Let R be the ﬁnest equivalence relation on T for which (y, t) ≡ (¯y, 1 − t), (y, 0) ≡ o(y), and

(y, 1) ≡ t(y) for all y ∈ E and t ∈ [0, 1]. Then the quotient space real(Γ) = T/R is called the realization of the graph Y .

We can deﬁne two graphs Pathn and Circn for n ≥ 1 in the following way. Let X =

{0, 1, . . . , n}, and let E+ = {y1, . . . , yn} be the set of edges satisfying o(yi) = i − 1, t(yi) = i; a ¯ 0 then X = vert Pathn and E = E+ E+ = edge Pathn. If we let X = {0, 1, . . . , n − 1} and

0 0 0 0 0 let E+ = {y1, . . . , yn}, where o(yn) = n − 1 and t(yn) = 0 and the other extremities are as 0 0 0 a ¯0 in Pathn, then X = vert Circn and E = E+ E+ = edge Circn. The realizations of these two graphs are precisely what you would expect; a segment with n edges and a circuit with n edges, respectively.

Deﬁnition. A path of length n in a graph Y is an injective morphism of Pathn into Y ; a circuit of length n in Y is any subgraph of Y isomorphic to Circn. A nonempty connected graph without circuits is called a tree.

The idea is to try to understand the fundamental group of a connected, closed, oriented

3-manifold by constructing a graph on which the group acts, then using knowledge of graphs to understand the fundamental group.

Deﬁnition. A graph of groups (G, Y ) consists of a graph Y , a group GP for each P ∈ vert Y , and a group Gy for each y ∈ edge Y , together with a monomorphism Gy → Gt(y) (denote 38 y y the image of an element a 7→ a and denote by Gy the image of Gy in Gt(y)); one requires in addition that Gy¯ = Gy.

When thought of as the realization of a Serre-type graph, a graph of groups is simply an oriented graph, with groups assigned to each vertex and edge in such a way that the edge groups inject into the vertex groups at either end of the edge. The most important example is the segment:

Gy GP GQ

Implied here are monomorphisms Gy → GQ and Gy → GP . What we would like to compute is the fundamental group of the graph of groups. To deﬁne and compute this will require some machinery.

Let (G, Y ) be a graph of groups. Let Γ be the free product of the GP for each P ∈ vert Y and the free group with basis y ∈ edge Y . Deﬁne the group F (G, Y ) to be the quotient of

Γ by the normal subgroup generated by elements of the form yy¯ and yayy−1(ay¯)−1 for all

−1 y ∈ edge Y , a ∈ Gy; that is, F (G, Y ) is the group Γ with relations of the form y =y ¯,

y −1 y¯ ya y = a imposed. Let c be a path of length n in Y . Let y1, . . . , yn be the edges of c, and put Pi = o(yi+1) = t(yi).

Deﬁnition. A word of type c in F (G, Y ) s a pair (c, µ) where µ = (r0, . . . , rn) is a sequence of

elements ri ∈ GPi . The element |c, µ| = r0y1r1y2 ··· ynrn of F (G, Y ) is said to be associated with the word (c, µ).

Note that we do not bother to distinguish between the element of a group GP and its image GP → F (G, Y ); this abuse of notation is ﬁne, as the maps are injective (see [Ser03, Corollary 1]). We now give two distinct deﬁnitions of the fundamental group of the graph of groups, then claim that they are isomorphic.

39 Deﬁnition. Let (G, Y ) be a graph of groups, and let P0 be a vertex of Y . We call the fundamental group of (G, Y ) at P0 the subgroup π1(G, Y, P0) of F (G, Y ) consisting of elements of the form |c, µ|, where c is a path in Y that starts and ends at P0.

∗ ∗ Alternatively, let Y be a maximal tree of Y . The fundamental group π1(G, Y, Y ) of (G, Y ) at Y ∗ is the quotient of F (G, Y ) by the normal subgroup generated by elements

y ∈ edge Y ∗.

∗ One could think of the group π1(G, Y, Y ) as having the same generators and relations as F (G, Y ), with the additional relations y = 1 if y ∈ edge Y ∗. In particular, the segment

Gy GP GQ

has fundamental group π1(G, Y, Y ) = GP ∗Gy GQ.

Proposition 1.53 ([Ser03, Proposition 20]). Let (G, Y ) be a graph of groups, let P0 ∈ vert Y

∗ ∗ and let Y be a maximal tree of Y . The canonical projection p : F (G, Y ) → π1(G, Y, Y )

∗ induces an isomorphism of π1(G, Y, P0) onto π1(G, Y, Y ).

In particular, this shows that the fundamental group is independent of the choice of

∗ vertex P0 or maximal tree Y , much as the fundamental group of a topological space is independent of the choice of base point.

Let us now consider the elements of a fundamental group of a graph of groups.

Deﬁnition. We say that a word (c, µ) is reduced:

• If n = 0, r0 6= 1;

• If n ≥ 1, then r ∈/ Gyi for each i such that y =y ¯ . i yi i+1 i

This deﬁnition essentially requires that no backtracking is allowed in the path c unless

something of interest is done in the midst of backtracking; in particular, one must have

attached an element ri ∈ GPi at the vertex Pi not in the image of the group Gyi−1 associated

with the edge yi−1 used to get there. 40 Theorem 1.54 ([Ser03, Theorem 11]). If (c, µ) is a reduced word, the associated element

|c, µ| of F (G, Y ) is distinct from the identity.

Corollary 1.55 ([Ser03, Corollary 3]). Let Y ∗ be a maximal tree of Y , and let (c, µ) be a

∗ reduced word whose type c is a closed path. Then the image of |c, µ| in π1(G, Y, Y ) is distinct from the identity.

We now wish to deﬁne the Bass-Serre tree of a group of graphs.

Deﬁnition. Let Y be a graph on which a group G acts. An inversion is an element g ∈ G and an edge y ∈ edge Y such that gy =y ¯. If no such pair exists, we say that G acts without inversion; equivalently, G acts without inversion if there is an orientation Y+ preserved by G.

If G acts on Y without inversion, we can deﬁne the quotient graph G\Y simply by taking the quotient of edge Y and vert Y with respect to their respective induced G-actions; if G had an inversion, the induced map y 7→ y¯ would fail to satisfyy ¯ 6= y.

Let us now suppose we have a graph of groups (G, Y ), a maximal tree Y ∗ in Y , and an orientation A of Y . For each y ∈ edge Y , deﬁne |y| to be the unique edge in {y, y¯} ∩ A. Let

∗ π = π1(G, Y, Y ) be the fundamental group of the graph of groups. We wish to construct a graph T = T (G, Y, Y ∗) and a π-action on T without inversion so that T is a cover of Y and the projection map p : T → Y induces a graph isomorphism π\T → Y . a Let vert T = π/GP be the disjoint union over the vertices of Y of left cosets of P ∈vert Y GP (or more appropriately, the isomorphic image of GP in π). Then π has a natural action a |y| on vert T , namely that of left multiplication. Similarly, let edge T = π/G , or the |y| y∈edge Y disjoint union over the edges of Y of the isomorphic image of the group Gy in the group

y Go(|y|) (recall that Gy is the image of Gy in the terminal vertex group GP ,P = t(y)). Again, π has the usual action on T by left multiplication. It is also clear what the projection map p must be; for each P ∈ vert Y , p(π/G ) ≡ P and for each y ∈ edge Y , p π/G|y| ≡ y. p |y| 41 In order to make T a graph, we need to specify the inverse and extremities of each edge.

First, we identify a princpal lift of Y in T (or equivalently, a section of p). To this end, let

P = G ∈ vert T and let y = G|y| ∈ edge T ; that is, each vertex or edge is identiﬁed with e P e |y| its corresponding coset that contains 1 ∈ π. Inverses are simple; for each edge y ∈ edge Y

∗ we want ye = ey¯. For the extremities, suppose ﬁrst that y ∈ A. Recall that π(G, Y, Y )

is a quotient of the group F (G, Y ); let gy be the image of the edge y ∈ F (G, Y ) under

the canonical projection. Then we set o(ye) = og(y) to be the principal lift of the vertex

P = o(y) in Y , and we set t(ye) = gytg(y) to be the coset in π/Go(y) containing gy. Note that if y ∈ edge Y ∗, then y is a generator of the normal group by which we quotient F (G, Y ) to

obtain π; in that case, gy = 1 and t(ye) = tg(y). On the other hand, we expect T to be a tree even if Y is not, so it would be unwise to expect to ﬁnd a copy of Y sitting in T ; instead, we

will obtain a copy of Y in which each cycle has has one edge disconnected at the terminal

vertex P and connected to a distinct vertex in the p-preimage of P (in particular, each edge

∗ not in edge Y ). We can now deﬁne o(ey¯) and t(ey¯) consistently with the extremities of ye to extend our deﬁnitions to the full section.

To extend the choices to all of edge T , we now rely on the π action. For any g ∈ π, let

gye = gey¯, o(gye) = g · og(y), and t(gye) = ggy · tg(y) for y ∈ A, and define it consistently on each inversey ¯. Note that these definitions are well-defined; let gy = g0y. Then g−1g0 ∈ G|y| ≤ e e |y|

Go(|y|); this implies that the extremities are well-deﬁned. Also Gy = Gy¯, so cosets of their

image in Go(|y|) correspond. We then have the following result:

Theorem 1.56 ([Ser03, Theorem 12]). Let (G, Y ) be a connected, nonempty graph of groups,

let Y ∗ be a maximal tree of Y and let A be an orientation of Y . Then the graph T constructed

above is a tree.

Moreover, if X is a tree and G is a group acting on X without inversion, then we can

construct a graph of groups (G, Y ) with universal cover T isomorphic to X and G isomorphic

42 to the fundamental group of (G, Y ) (see [Ser03, Section I.5.4]). In particular, we have the following result:

Theorem 1.57 ([Ser03, Theorem 13]). Let X be a connected nonempty graph and let G be a group acting on X without inversion. Let Y = G\X be the quotient of X by the G-action, let

Y ∗ be a maximal tree in Y , and let A be an orientation of Y . Let (G, Y ) be the corresponding graph of groups, and let T = T (G, Y, Y ∗) be the universal cover of (G, Y ). Then the following are equivalent:

• X is a tree;

• T is isomorphic to X;

∗ ∼ • π1(G, Y, Y ) = G.

By this result, the choice of maximal tree and orientation is irrelevant; we call T the

Bass-Serre covering tree of the graph of groups (G, Y ). It is of particular use in constructing models for a manifold in terms of models for the pieces in a decomposition.

43 CHAPTER 2

VIRTUALLY CYCLIC MODEL CONSTRUCTION METHODS

2.1 Virtually Cyclic Extensions

Let Γ be a ﬁnitely generated discrete group that surjects onto the group Γ0 via the group homomorphism φ. Let F be a family of subgroups of Γ. We wish to use knowledge of Γ0 to construct a model for the classifying space EF Γ.

Deﬁnition. Let Γ be any ﬁnitely generated group, and F ⊂ F 0 a pair of families of subgroups

0 of Γ. We say that a collection A = {Aα}α∈I of subgroups of Γ is adapted to the pair (F, F ) provided that;

1. For all A, B ∈ A, either A = B or A ∩ B ∈ F;

2. The collection A is conjugacy closed; i.e., if A ∈ A then g−1Ag ∈ A for all g ∈ Γ;

3. Every A ∈ A is self-normalizing; i.e., NΓ(A) = A;

4. For all A ∈ F 0 \F, there exists B ∈ A such that A ≤ B.

The usefulness of the adapted collection is that it allows you to build a model for EF 0 Γ

0 from a model for EF Γ. Let Γ be a discrete group, let F ⊆ F be families of subgroups of Γ, let A be a collection adapted to the pair (F, F 0). We review how to construct a model for

EF 0 Γ; a full discussion and proof can be found in [LO09a, p. 302].

44 Proposition 2.1. Let F ⊂ F 0 be a nested pair of families of subgroups of Γ, and assume

0 that the collection of subgroups A = {Hα}α∈I is adapted to the pair (F, F ). Let H be a complete set of representatives of the conjugacy classes within A, and consider the cellular

Γ-pushout:

a β Γ ×H EF H EF Γ H∈H

α a Γ ×H EF 0 H X H∈H

Then X is a model for EF 0 Γ. In the above cellular Γ-pushout, we require either (1) α is the disjoint union of cellular H-maps (H ∈ H), β is an inclusion of Γ-CW-complexes, or (2)

α is the disjoint union of inclusions of H-CW-complexes (H ∈ H), β is a cellular Γ-map.

Next we consider how to “transfer” adapted collections between diﬀerent pairs of families:

Lemma 2.2. Let Γ be a ﬁnitely-generated discrete group, and let F ⊆ F 0 ⊆ F 00 be three

nested families of subgroups of Γ. Let A be a collection adapted to the pair (F, F 00); then A is adapted to the pairs (F, F 0) and (F 0, F 00) as well.

Proof. We check the four requirements of adaptation directly for the pairs (F, F 0) and

(F 0, F 00).

1. Let A, B ∈ A and suppose A 6= B. Then since A is adapted to (F, F 00), A∩B ∈ F ⊆ F 0

and the ﬁrst condition holds for both pairs.

2. The second condition (closure under conjugacy) is independent of the family pair, and

thus carries through.

3. The third condition (self-normalizing) is also independent of the family pair.

45 4. Let A ∈ F 0 \F; then A ∈ F 00 ⊇ F 0. Similarly, if A ∈ F 00 \F 0, then A/∈ F ⊆ F 0. In

either case A ∈ F 00 \F, so there exists an B ∈ A with A ≤ B.

Lemma 2.3. Let φ :Γ → Γ0 be a homomorphism between ﬁnitely generated groups, let

0 F ⊆ F be a pair of families of subgroups of Γ0, and let A = {Aα}α∈I be a collection adapted

0 −1 0 to the pair (F, F ). Then Ae = {φ (Aα)}α∈I is a collection adapted to the pair (Fe, Fe ) of families of subgroups of Γ.

Proof. We check the four conditions directly.

1. Let A,e Be ∈ Ae; then there exist A, B ∈ A with Ae = φ−1(A) and Be = φ−1(B). Since A is an adapted collection, we must have either A = B or A ∩ B ∈ F. If A = B, then

Ae = φ−1(A) = φ−1(B) = Be. If A 6= B then

φ(Ae ∩ Be) ≤ φ(Ae) ∩ φ(Be) = A ∩ B ∈ F,

and since F is closed under passing to subgroups we have that φ(Ae∩ Be) ∈ F and thus

that Ae ∩ Be ∈ Fe.

2. Let Ae ∈ Ae, g ∈ Γ; we want g−1Age ∈ Ae. Let

−1 −1 A = φ(Ae) ∈ A, h = φ(g) ∈ Γ0 so that φ(g Age ) = h Ah.

A is an adapted collection, so we have h−1Ah ∈ A; let Be = φ−1(h−1Ah) ∈ Ae. Certainly

g−1Age ⊆ Be; let b ∈ Be. Then φ(b) ∈ h−1Ah, so there exists a ∈ A such that φ(b) =

h−1ah, or a = φ(gbg−1). Then gbg−1 ∈ φ−1(φ(Ae)) = Ae, so b ∈ g−1Age . Thus g−1Age =

Be ∈ Ae, as desired.

3. Let Ae ∈ Ae; we’d like to show that NΓ(Ae) = Ae. Certainly Ae ≤ NΓ(Ae); let g ∈ NΓ(Ae).

Then g−1Age ≤ Ae, so φ(g−1Age ) = h−1Ah ≤ A = φ(Ae). But A is a collection of self-

normalizing subgroups, so h ∈ NΓ0 (A) = A, and g ∈ Ae. Thus, Ae is self-normalizing. 46 4. Let Ae ∈ Fe0 \ Fe; then φ(Ae) = A ∈ F 0 \F, so there exists B ∈ A such that A ≤ B.

Then Ae ≤ φ−1(φ(Ae)) = φ−1(A) ≤ φ−1(B) ∈ Ae, as desired.

Now we have a method to construct a model for the classifying space EF Γ.

Theorem 2.4. Let F be a family of subgroups of the ﬁnitely-generated discrete group Γ. Let

0 φ :Γ → Γ0 be a homomorphism. Let F0 ⊆ F0 be a nested pair of families of subgroups of Γ0

0 0 satisfying Fe0 ⊆ F ⊆ Ff0, and let A = {Aα}α∈I be a collection adapted to the pair F0 ⊆ F0.

−1 Let H be a complete set of representatives of the conjugacy classes within Ae = {φ (Aα)}α∈I , and consider the following cellular Γ-pushout:

a β Γ × E H E Γ He F0 F0 0 He∈H

α a Γ × E H X He F e He∈H

Then X is a model for EF Γ. In the above cellular Γ-pushout, we require either (1) α is

the disjoint union of cellular He-maps (He ∈ H), β is an inclusion of Γ-CW-complexes, or (2)

α is the disjoint union of inclusions of He-CW-complexes (He ∈ H), β is a cellular Γ-map.

Proof. From Proposition 1.3 we have that EF Γ0 = E Γ and EF H = E H. We know 0 Ff0 0 Ff0 e

from Lemmas 2.2 and 2.3 that Ae is a collection adapted to the pair (Ff0, F). Then by

Proposition 2.1 we have that the above pushout is a model for EF Γ.

In order to use the above contruction, one must ﬁrst have knowledge of EF0 Γ0, Ae, EF0 H and EF He. We are particulaly interested in constructing models for EΓ where Γ is an extension of

Γ0 by a group K

φ 0 → K → Γ → Γ0 → 0 47 where K is well understood; consider the case where K is virtually cyclic.

Given a model for EΓ0, a collection A adapted to the pair of families of ﬁnite and

virtually cyclic subgroups in Γ0 and models for EH and EHe for He ∈ Ae, we have the following construction for EΓ:

Corollary 2.5. Let Γ be a ﬁnitely-generated discrete group. Let φ :Γ → Γ0 be a surjective

homomorphism with virtually cyclic kernel. Let FIN 0 and VC0 be the families of ﬁnite

and virtually cyclic subgroups of Γ0, respectively. Let A be a collection adapted to the pair

(FIN 0, VC0). Let H be a complete set of representatives of the conjugacy classes within Ae, and consider the following cellular Γ-pushout:

a Γ × EH EΓ He 0 He∈H

a Γ × EH X. He e He∈H

Then X is a model for EΓ. In the above cellular Γ-pushout, we require either (1) α is the

disjoint union of cellular He-maps (He ∈ H), β is an inclusion of Γ-CW-complexes, or (2) α is the disjoint union of inclusions of He-CW-complexes (He ∈ H), β is a cellular Γ-map.

Proof. The φ-preimage of all ﬁnite groups in Γ0 are clearly virtually cyclic, and elements of VC must map to virtually cyclic subgroups in Γ, so FIN^0 ⊆ VC ⊆ VCg0. Applying Theorem 2.4 gives the result.

Having constructed a d-dimensional model for EF Γ gives us an upper bound; gdF (Γ) ≤ d. In order to get a lower bound, we can often demonstrate that Γ has a subgroup H for which we know gdF (H) = d, as this forces gdF (Γ) = d by Corollary 1.4. If F is the family of ﬁnite ∼ n n or virtually cyclic subgroups we often ﬁnd a subgroup H = Z , as we know gd(Z ) = n and n gd(Z ) = n + 1 by Proposition 1.48 and Theorem 1.50. 48 If not, it may be necessary to work directly with the Bredon cohomology groups them-

d selves, demonstrating either that HF (Γ; M) is nontrivial for a given M ∈ Mod-OF Γ (usually

i M = Z), or that HF (Γ; M) is trivial for all i > gdF (Γ) and for all M ∈ Mod-OF Γ. For our pushout construction, this is most easily done by putting the Bredon cohomology of the models of the pushout into a long exact sequence called a Mayer-Vietoris sequence.

In particular for our pushout model, we have the following sequence ([LW12]):

Proposition 2.6. Let F be a family of subgroups of the ﬁnitely-generated discrete group Γ.

0 Let φ :Γ → Γ0 be a homomorphism. Let F0 ⊆ F0 be a nested pair of families of subgroups

0 0 of Γ0 satisfying Fe0 ⊆ F ⊆ Ff0, and let A be a collection adapted to the pair F0 ⊆ F0. Let H be a complete set of representatives of the conjugacy classes within Ae. Let M ∈ Mod-OF Γ. Then there exists a long exact cohomology sequence i Y i i Y i ... → H (Γ; M) → H (He; M) ⊕ H (Γ0; M) → H (He; M) F F F0 Ff0 He∈H He∈H

i+1 → HF (Γ; M) → ....

Remark 2.7. In general we will be after the case where F are the virtually cyclic subgroups

0 of Γ and F0 (resp. F0) are the ﬁnite (resp. virtually cyclic) subgroups of Γ0. In these cases, the Mayer-Vietoris sequence is written: i Y i i Y i ... → H (Γ; M) → H (He; M) ⊕ H (Γ0; M) → H (He; M) FIN^0 He∈H He∈H

→ Hi+1(Γ; M) → ....

Degrijse and Petrosyan give a complete construction of this sequence in [DP12, Section

7], in fact generalizing it to arbitrary Ext and Tor functors. The careful reader may have

noticed that, in the above Mayer-Vietoris sequence, the orbit module M ∈ Mod-OF Γ was

being used as an orbit module in Mod-OF He, Mod-OF0 Γ0, and Mod-OF0 H. This abuse of notation is justified by the following definition. 49 Definition. Let G and K be groups, F and V families of subgroups of the respective groups. Let π : OV K → OF G be a functor. Then there exist functors resπ : Mod-OV K →

Mod-OF G and indπ : Mod-OF G → Mod-OV K—called a restriction and induction functor, respectively—with

M resπ(M) = M ◦ π, indπ(M) = M(K/V ) ⊗ Z[Mor(−, π(K/V ))] /I. V ∈V For a given H ∈ F, the abelian group I is generated by elements of the form M(φ)(m) ⊗ n − m ⊗ N(φ)(n), where for some V,W ∈ V; φ ∈ Mor(K/V, K/W ), m ∈ M(K/W ), and n ∈ N(K/V ). Here N ∈ OV K-Mod is the right OV K module that takes the coset K/V to the free abelian group Z[Mor(G/H, π(K/V ))] generated by morphisms in Mor(G/H, π(K/V )).

Finally, under these deﬁnitions, indπ is left adjoint to resπ; that is,

∼ HomOF G(indπ(M),N) = HomOV K (M, resπ(N)).

Though the deﬁnitions are tedious, for our purposes the only need we have for the induction and restriction functors is the following lemma ([DP12, Lemma 7.5]):

Lemma 2.8. The induction and restriction functors associated to the functors

•O Γ → OF Γ:Γ/F → Γ/F Ff0

•O F He → OF Γ: H/Fe → Γ/F

•O H → OF Γ: H/F0 → Γ/F Ff0 e are all exact and preserve projectives.

This lemma not only allows us to abuse notation in the Mayer-Vietoris sequence for

Bredon cohomology by leaving oﬀ the restriction functors, but also allows us to get a module M ∈ Mod-OF Γ given a module in one of the other orbit categories. For example, if we know there is a module M ∈ Mod-OF| H with respect to which a particular Bredon He e

50 i cohomology group HF| (He; M) is nontrivial, then we also know that there is a module He ¯ M ∈ Mod-OF Γ that restricts to M under the appropropriate restrictor function resπ by ¯ taking M = indπ(M).

2.2 Acylindrical Splittings

Another useful construction is a generalization of a pushout construction of Lafont and

Ortiz ([LO09b]). This is another use of the pushout construction of L¨uck and Weiermann in

Proposition 2.1, this time speciﬁed to the case in which G has an acylindrical splitting. This is a generalization by Delzant ([Del99]) of an idea originally given by Sela ([Sel97]), related to the latter’s work on the accessibility problem for ﬁnitely generated groups.

Deﬁnition. Let G be a group, (G, Y ) a graph of groups. G is said to be acylindrical if there exists an integer k such that, for every path c of length k in the Bass-Serre tree T associated to the splitting of G, the stabilizer of c is ﬁnite.

In preparation of the construction, we quote two results from [LO09b]. The proofs of these claims generalize immediately to the situation given, as they are a function of the action of the fundamental group of a graph of groups on the Bass-Serre covering tree.

Lemma 2.9 ([LO09b, Claim 2]). Let (G, Y ) be a graph of groups where the splitting of G is acylindrical. Then the stabilizer of any geodesic in the Bass-Serre tree is a virtually cyclic subgroup of G. Furthermore, every virtually cyclic subgroup V ≤ G satisﬁng V ∈ VC \ F stabilizes a unique geodesic in T .

Lemma 2.10 ([LO09b, Claim 3]). Let (G, Y ) be a graph of groups where the splitting of G is acylindrical. Let F be the family of virtually cyclic subgroups of G that conjugate into one of the vertex groups, and let VC be the full family of virtually cyclic subgroups. Then the collection A = {Aα}α∈I of subgroups of G consisting of all maximal virtually cyclic subgroups of VC\F is adapted to the pair (F, VC). 51 Proposition 2.11. Let G be a ﬁnitely-generated discrete group. Let (G, Y ) be a graph of

groups in which the splitting of G is acylindrical. Let F be the family of virtually cyclic

subgroups of G that conjugate into a vertex group in (G, Y ). Let A be the collection of

maximal virtually cyclic subgroups of G not in F. Let H be a complete set of representatives

of the conjugacy classes within A. Let {∗} be the CW -complex that is a single 0-cell, and

consider the following cellular G-pushout:

a G ×H E EF G H∈H

a G ×H {∗} X. H∈H

Then X is a model for EG. In the above cellular Γ-pushout, we require either (1) α is

the disjoint union of cellular He-maps (He ∈ H), β is an inclusion of Γ-CW-complexes, or (2)

α is the disjoint union of inclusions of He-CW-complexes (He ∈ H), β is a cellular Γ-map.

Proof. By Lemma 2.10 and Proposition 2.1, we will know X is a model for EG if we can demonstrate that for any H ∈ H, E and {∗} are models for EF H and EH, respectively. Let H ∈ H; then H is virtually cyclic. This immediately allows {∗} to be a model for

EH, as the family of virtually cyclics in H is simply the full family of subgroups of H. It

only remains to show that E is a model for EF H.

First, we demonstrate that F|H is just the family of ﬁnite subgroups of H. Let V ∈ F

−1 satisfy V ≤ H. V can be conjugated into GP for some vertex group in (G, Y ); say gV g ≤

GP . V is also virtually cyclic; let hvi be a ﬁnite-index subgroup of V for some v ∈ V . If hvi

n is not ﬁnite, then [H : hvi] < ∞. Let h ∈ H \ hvi; then h ∈ hvi for some n ∈ N; this means n −1 that gh g ∈ GP . But GP is a generating set of the free product F (G, Y ) of vertex groups

and edges y ∈ edge Y ; as such, there is no element f ∈ F (G, Y ) \ GP with a power in GP .

−1 −1 Thus, it must be that ghg ∈ GP . But this implies that gHg ≤ GP , contradicting that

52 H/∈ F; thus, it must be that hvi is finite. So V has a finite subgroup of finite index, and

is therefore ﬁnite. On the other hand, if F ≤ H is a ﬁnite subgroup, then T F is nonempty

(note that T is CAT(0), so we apply Corollary 1.19). in particular, F ﬁxes some vertex

gGP ∈ vert T , and is therefore conjugate to a subgroup of GP ; in other words, F ∈ F. With this in mind, we are now after a model for EH. By Lemma 2.9, any H ∈ H ⊆ VC\F is the stabilizer in G of a unique geodesic c in T . As such, we can put H in the short exact sequence

φ 1 → Gc → H → Q → 1, where Gc is the subgroup of H that fixes c pointwise. Giving c the usual metric we have ∼ that c ' E; thus, the quotient Q = H/Gc inherits an effective action on E, and is therefore 1-crystallographic. Moreover, by Proposition 1.3, a model for EQ is a model for E H, FIN^ where FIN] is the family generated by φ-preimages of finite subgroups of Q. But because

G is acylindrical, and since c has paths of arbirtarily long length, Gc must be ﬁnite. Thus, FIN] is just the family of ﬁnite subgroups of H, and a model for EQ is therefore a model for

EH. By Proposition 1.48, E is a model for EQ and therefore for EH; the result follows.

Remark 2.12. As in Proposition 2.6, we have the following long exact sequence related to

the above pushout construction: i Y i i Y i ... → H (Γ; M) → H (H; M) ⊕ HF (Γ; M) → HF (H; M) H∈H H∈H

→ Hi+1(Γ; M) → ....

In this particular case, since we know that cd(H) ≤ gd(H) = 1 and cd(H) ≤ gd(H) = 0, for

each i > 2 we have that

i i 0 → H (Γ; M) → HF (Γ; M) → 0

i ∼ i is exact; in particular, H (Γ; M) = HF (Γ; M) for each i > 2, M ∈ Mod-OΓ.

53 CHAPTER 3

VIRTUALLY CYCLIC MODELS FOR GEOMETRIC GROUPS

3.1 Seifert Fiber Spaces

We begin by giving the definition of a Seifert fiber space. The given definition and results,

as well as a more detailed discussion, can be found in [Sco83].

Definition. A trivial fibered solid torus is the usual product S1 × D2 with the product foliation by circles S1 × {y}, y ∈ D2.A fibered solid torus is a solid torus with a foliation by circles which is finitely covered by a trivial fibered solid torus. Similarly, a fibered solid

Klein bottle is a solid Klein bottle which is ﬁnitely covered by a trivial solid ﬁbered torus.

Remark 3.1. Nontrivial ﬁbered solid tori can be constructed by cutting a trivial one along any cross section {x}×D2, twisting by p/q of a full turn (p, q > 0 coprime), then gluing back

together. The resulting ﬁbered solid torus is q-covered by a trivial ﬁbered solid torus. The

ﬁbers in the newly constructed solid torus are all—with the exception of the central ﬁber—

loops that wind around the body of the torus q times. The central ﬁber is thus called a critical

ﬁber; the rest are regular ﬁbers.

A fibered solid Klein bottle is constructed by cutting a trivial fibered solid torus, reflecting

one end through any line through the center of D2, and gluing the ends back together. All

fibers through the reflection line are critical, while the remaining fibers are regular.

Deﬁnition. A Seifert ﬁber space is a 3-manifold with a decomposition into disjoint circles,

54 called fibers, such that each circle has a neighborhood which is a union of fibers and is isomorphic to a fibered solid torus or fibered Klein bottle.

Given a Seifert fiber space M, one can obtain a quotient space B by quotienting out by the S1-action on the fibers of M; that is, by identifying each fiber to a point. By considering the quotient of neighborhoods of fibers in M, the topology B inherits makes it a surface with a natural orbifold structure; we call B the base orbifold of M.

The following result gives us the exact sequence needed to apply the method of the previous chapter ([Sco83, Lemma 3.2]):

Lemma 3.2. Let M be a Seifert ﬁber space with base orbifold B. Let Γ be the fundamental group of M, and let Γ0 be the orbifold fundamental group of B. Then there is an exact sequence

1 → K → Γ → Γ0 → 1, where K denotes the cyclic subgroup of Γ generated by a regular ﬁber. The group K is inﬁnite except in cases where M is covered by S3.

Note that B is not generally a manifold, as the action of S1 on M by which we quotient out is not generally free; in particular, critical fibers are fixed by nontrivial elements of S1, leading to singularities in the base space. A single critical fiber in a nontrivial fibered solid torus gives rise to a cone point, while fibers through a reflection line in the construction of a

ﬁbered solid Klein bottle give rise to reﬂector curves. These names follow from considering

2 the quotient of E by either a finite group of rotations about the origin or a reflection about 2 a line. In the former case the quotient is homeomorphic to E , but the inherited metric 2 from E behaves differently at the image of the origin; as such, is it more helpful to think of this quotient as a cone with angle 2π/n, with n being the number of rotations in the group.

The latter case is a half-space. The combination of the two—achieved by quotienting out

55 by a ﬁnite group of rotations and a reﬂection about a line through the origin—has a corner

reﬂector as the image of the origin.

Though the base orbifold of a closed manifold may well have a boundary as a topological

space, we deﬁne the boundary of an orbifold slightly diﬀerently, as per Scott ([Sco83, Section

2]).

Deﬁnition. An n-orbifold without boundary is a Hausdorﬀ, paracompact space which is

n locally homeomorphic to the quotient space of R by a ﬁnite group action. In other words, reﬂector curves do not count as part of the orbifold boundary.

From now on, we refer to an orbifold without boundary in this sense. It is clear that

the base orbifold B of a Seifert ﬁber space M has nonempty boundary if and only if M has

nonempty boundary in the usual sense.

As our objective is to compute gd(Γ) for fundamental groups of connected, closed, ori-

ented three manifolds, we assume that M is connected and oriented; we will still need to

allow M to have nonempty boundary when considering decompositions. As M is oriented,

all neighborhoods of ﬁbers in M are isomorphic to a ﬁbered solid torus; cone points are

therefore the only critical points in the base orbifold B.

Now suppose M is closed.

Deﬁnition. An orbifold is called good if it is covered by some manifold or, equivalently, if its universal covering orbifold is a manifold. An orbifold that is not good is called bad.

Simply knowing that a closed 2-orbifold is good is enough to tell you that it is ﬁnitely covered by a manifold:

Theorem 3.3 ([Sco83, Theorem 2.5]). Every good, compact 2-dimensional orbifold without boundary is ﬁnitely covered by a manifold.

We now begin to classify 2-orbifolds.

56 Theorem 3.4 ([Sco83, Theorem 2.4]). Every good 2-dimensional orbifold without boundary

2 2 2 admits a geometric structure modeled on S , E , or H ; that is, it is isomorphic as an orbifold 2 2 2 to the quotient of S , E , or H by some discrete group of isometries.

Theorem 3.5 ([Sco83, Theorem 2.3]). The only bad 2-orbifolds without boundary are of the following types:

• A sphere S2 with a single cone point (the teardrop orbifold);

• A sphere S2 with two cone points of diﬀerent angles;

• A disc D2 whose boundary is formed by a corner reﬂector with the ends of the reﬂector

curves identiﬁed;

• A disc D2 whose boundary is formed by two corner reﬂectors of diﬀerent angles, with

each end of one corner reflector identified with a different end of the other corner

reﬂector.

The bad 2-orbifolds by deﬁnition can’t be modeled on a two-dimensional geometry; they can still be dealt with simply by ﬁrst generalizing the Euler characteristic from compact manifolds to compact orbifolds. This is done using the Riemann-Hurwitz formula (sometimes called Hurwitz’s Theorem; see [Har77, Section IV.2] or the discussion in [Sco83, pp.426-7]),˜ presented here in a form more directly useful for our computations.

Theorem 3.6 (Hurwitz). Let B be a compact orbifold, Y its underlying surface. Suppose

B has m cone points with cone angle 2π/qi, 1 ≤ i ≤ m, and n corner reﬂectors with angle

π/rj, 1 ≤ j ≤ n. Then the Euler characteristic of B is given by the formula

m n X 1 1 X 1 χ(B) = χ(Y ) − 1 − − 1 − . q 2 r i=1 i j=1 j Scott uses this invariant to classify closed Seifert ﬁber bundles.

57 Theorem 3.7 ([Sco83, Theorem 5.3(ii)]). Let M be a closed Seifert ﬁber space with base

3 3 2 orbifold B. Then M possesses a geometric structure modeled on one of S , E , S × E, 2 H × E, PSLg 2(R) or Nil. In particular, we have the following classiﬁcation based on χ(B):

2 3 • If χ(B) > 0 then M is modeled on S × E or S ;

3 • If χ(B) = 0 then M is modeled on E or Nil;

2 • If χ(B) < 0 then M is modeled on H × E or PSLg 2(R).

The choice in each case depends upon whether M has a ﬁnite covering which is a trivial circle bundle over a surface; if M possesses such a covering, then M is modeled on one of

2 3 2 S × E, E or H × E.

Remark 3.8. In [Sco83] two invariants are used to classify M: χ(B) and e(η) (the Euler number), where η : M → B is the ﬁbration map. Scott gives an earlier lemma [Sco83,

Lemma 3.7] that gives a geometric interpretation of e(η), which is the interpretation used above.

With this preparation, we can now begin to demonstrate results.

Proposition 3.9. Let M be a closed Seifert ﬁber space with base orbifold B and fundamental

2 2 group Γ. If Γ is not virtually cyclic, then B is a good orbifold modeled on either E or H .

2 Proof. We proceed by contraposition; suppose ﬁrst that B is modeled on S . Then Γ0 = ∼ 2 π1(B) is a discrete subgroup of SO(3) = Isom(S ) and is therefore ﬁnite; by the short exact sequence given in Lemma 3.2, Γ is virtually cyclic.

Similarly, if B is a bad manifold then by Theorem 3.5, χ(B) > 0. Thus by Theorem 3.7, 3 2 ∼ 3 M is modeled on one of S or S × E; discrete subgroups in either SO(4) = Isom(S ) or ∼ 2 SO(3) × (R o Z2) = Isom(S × E) are virtually cyclic.

58 Since we know a point is a model for EΓ if Γ is virtually cyclic, we know immediately

that gd(Γ) = 0 in these cases. We are left with three cases:

1. M has nonempty boundary;

2 2. B is modeled on H ;

2 3. B is modeled on E .

3.1.1 Compact base orbifolds with nonempty boundary

Let M be a compact Seifert ﬁbered manifold with nonempty boundary. let Γ = π1(M) be

orb the fundamental group, let Γ0 = π1 (B) be the fundamental group of the base orbifold B,

and let φ :Γ → Γ0 be the associated homomorphism. We discuss what sort of orbifolds B can arise.

◦ Note ﬁrst that B is a good orbifold with fundamental group Γ0; one could consider the interior M ◦ of M and let B◦ be the base; then M ◦ doesn’t have boundary, but is also

not compact. By Theorem 3.5, the only bad orbifolds without boundary have compact

underlying space, so B◦ must be good, and therefore ﬁnitely covered by a 2-manifold N ◦

that is also not compact. Then N ◦ is geometric; by the uniformization theorem, all geometric

2 2 2 2 surfaces are modeled on S , E , or H . S is compact, so quotients by discrete (finite) actions ◦ 2 2 are also compact. As N is not compact, it must therefore be modeled on either E or H ; ◦ 2 2 B is therefore also modeled on one of E or H . ◦ 2 ◦ ◦ Suppose first that B is modeled on E . We obtain B via a quotient N /G, where ◦ ◦ G is a finite group acting on N . Similarly, N is a quotient of E by a group A, where 2 2 A ≤ Isom(E ) acts freely and discretely on E . This forces A to be a discrete group of ∼ k ∼ 2 ◦ translations, meaning A = Z with k either 0, 1, or 2. If A = Z then N ' T is the torus, ∼ contradicting that it is not compact. If A is trivial, then Γ0 = G is finite, Γ is virtually cyclic and the point space {∗} is a model for EΓ (in fact, this situation cannot arise in the

59 decompositions toward which we are aimed, as it cannot come from removing tori from a

2 closed, irreducible manifold; Z is not a subgroup of Γ). ∼ ◦ 1 1 ◦ Suppose A = Z; then N ' S × E. Letting N = S × [0, 1] so that N is the interior of N, we now have that B is a quotient of N by a ﬁnite action G. As we are largely concerned with orientable manifolds M, and as the Z-action on Mf is orientable, we may assume that B is itself orientable; thus, the only possible G-actions are the trivial action or rotation about an axis perpendicular to the [0, 1]-component (that is, a rotation that takes S1 × {0} to

1 S × {1}). 1 ∼ 2 If G is trivial, then B ' S × [0, 1], M ' T × [0, 1], Γ = Z and we have a model for EΓ 2 by Proposition 1.49. Moreover, we know gd(Z ) = 3 (Theorem 1.50). ∼ Suppose G = Z2 is the rotation action described above. Parametrize N by (z, t), with t ∈ [0, 1] and z ∈ C with |z| = 1. Then we can deﬁne σ : N → N to be the generator of A, where σ(z, t) = (¯z, 1 − t).

1 2 Consider the Seifert fiber space S × N ' T × [0, 1]; it has fundamental group Z and projects down via π onto N. Parametrize T × [0, 1] by (w, z, t), w, z ∈ C with |w| = |z| = 1 and t ∈ [0, 1]. We would like to define an orientation-preserving isometry σe on T × [0, 1] of order 2 with π ◦ σe = σ; the only possibility is σe(w, z, t) = (−w, z,¯ 1 − t). Consider the quotient M = T×[0, 1]/hσei. We have that M ' T×[0, 1]/ ∼, where the equivalence relation ∼ is defined by (w, z, t) ∼ (w0, z0, t0) if w0 = −w, z0 =z ¯, t0 = t = 0. Deformation retracting along [0, 1], we see that M deformation retracts down to a Klein bottle; this implies that 2 ∼ 2 M is modeled on E (with π1(M) = Z o Z2 a 2-crystallographic group). In particular, we again have a model for EΓ by Proposition 1.49 with gd(Γ) = 3 by Theorem 1.50. As such, we have shown the following:

Proposition 3.10. Let M be a compact Seifert ﬁbered manifold with nonempty boundary.

Let Γ = π1(M), and let B be the base orbifold of M. If B is not hyperbolic, then Γ is 2-crystallographic; in particular, gd(Γ) = 3.

60 Let us now assume that B is hyperbolic. We would like to use Proposition 2.1 to construct

a model for EΓ; we will do so in a way slightly diﬀerent from the construction of our other models, letting F 0 be the family of virtually cyclic subgroups of Γ and F be the family

of virtually cyclic subgroups of Γ satisfying φ(F ) ≤ Γ0 is ﬁnite for each f ∈ F (that is,

F = FIN^0).

In order to use Proposition 2.1, we are going to need a model for EF Γ, an adapted

collection A, and models for EF H and EH for each H ∈ A. We begin by constructing EF Γ. Let N be the ﬁnite manifold cover of B; then N has fundamental group that is free and

orb ﬁnitely generated by at least 2 isometries. We therefore have that π1 (B) is virtually free; this lets us use the following theorem ([KPS73]):

Theorem 3.11. Let G be a ﬁnite extension of a ﬁnitely generated virtually free group. Then

G is the fundamental group of a ﬁnite graph of ﬁnite groups (G, Y ); that is, vert Y and

edge Y are ﬁnite, as are GP and Gy for each P ∈ vert Y , y ∈ edge Y .

Let (Γ0,Y ) be the graph of groups given above, and let T be the Bass Serre covering

tree. Then Γ0 acts (without inversion) on T in the usual way.

Claim 3.12. Let Γ0 be a finite extension of a finitely generated free group, and let (Γ0,Y ) be an associated finite graph of finite groups. Let T be the Bass-Serre covering tree. Then T

is a model for EΓ0.

Proof. Suppose H ≤ Γ0 is not ﬁnite; then H is not conjugate to any subgroup of any (ﬁnite)

H H vertex subgroup, so T = ∅. Conversely, if H ≤ Γ0 is ﬁnite, then again T is nonempty

([Ser03, Example 6.3.1]). As Γ0 acts without inversion, H ﬁxes a subgraph of T ; suppose P and Q are two vertices of T in T H . Then there is a unique path in T between P and Q,

so H must ﬁx that path; in particular, T H is a connected subgraph of T . T H is therefore a

H tree, so T is contractible. We conclude that T is a model for EΓ0.

By applying Proposition 1.3, we see that T is a model for EF Γ. 61 Next, we demonstrate an adapted collection A.

Claim 3.13. Let Γ be the fundamental group of a compact Seifert ﬁbered manifold with nonempty boundary, and let Γ0 be the fundamental group of the hyperbolic base orbifold B. Then the collection A of subgroups that are preimages of maximal inﬁnite virtually cyclic

0 subgroups of Γ0 is adapted to the pair (F, F ) of families of subgroups of Γ.

Proof. Note first that virtually cyclic subgroups of Γ0 that conjugate into a vertex group of the graph of groups presentation must be finite, as the vertex groups themselves are finite; in particular, the splitting of Γ0 given by the graph of groups is acylindrical. By Lemma 2.10, the collection A0 of maximal infinite virtually cyclic subgroups of Γ0 is adapted to the pair

(FIN 0, VC0) of families of ﬁnite and virtually cyclic subgroups of Γ0, respectively. By

Lemma 2.3, A = Af0 is therefore adapted to the pair (F, VCg0) of families of subgroups of Γ.

0 0 Since F ⊆ F ⊆ VCg0, Lemma 2.2 gives that A is adapted to the pair (F, F ), as claimed.

Let He ∈ A be the φ-preimage of a maximal inﬁnite virtually cyclic subgroup H ≤ Γ0.

First, we construct a model for EF He; by Proposition 1.3, we use a model for EH.

Claim 3.14. If H ∈ A0 is a maximal inﬁnite virtually cyclic subgroup of Γ0, then E is a model for EH.

Proof. By Lemma 2.9, we know that H is the stabilizer subgroup of some geodesic c in T .

Let Hc ≤ H be the subgroup that fixes c pointwise; then since the splitting is acylindrical and this subgroup fixes paths of any particular length (in particular, vertexes), Γc must be ∼ finite. We can thus consider H as a finite extension of the quotient Q = H/Hc. But Q inherits an effective action on c ' E, so Q is 1-crystallographic and E is a model for EQ

(see Proposition 1.48). We ﬁnally pull up along the quotient map H → Q to get that E is a model for EH by Proposition 1.3.

It remains to construct a model for EHe. We will demonstrate that He has a quotient

62 Q by a ﬁnite group with Q 2-crystallographic, letting us use a model for EQ given by

Proposition 1.49 (again, the quotient map and Proposition 1.3 allow this).

Claim 3.15. Let He ∈ A as deﬁned above. Then H surjects onto a 2-crystallographic group Q, with ﬁnite kernel.

2 Proof. Consider H = φ(He) and its action on H (on which B is modeled). As H is virtually 2 cyclic, we know it either stabilizes a unique geodesic c or a unique boundary point ξ ∈ ∂H . 2 In the former case, H has a natural action on E as c ' E. In the latter case, ∂H \{ξ}' E (consider without loss of generality the case ξ = ∞ in the upper half-plane model), so H has

a natural action on E.

We now consider the preimage of this copy of E in the lift of the Seifert ﬁber space M 3 to its universal cover. By Lemma 3.2, Γ → Γ0 is inﬁnite cyclic unless M is modeled on S ,

when the supposition that M is not compact is contradicted; lifting E to Mf we then get a 2 2 He-action on E . Let Fe ≤ He be the subgroup with trivial action on E ; then F = Fe must be ﬁnite, as F ≤ H cannot contain the hyperbolic or parabolic element that generates the

finite-index infinite cyclic subgroup of H. Therefore Fe is itself finite, as it is an extension of ∼ 1 2 F by the (trivial) subgroup of ker φ = Z with trivial action above the S fiber in E . Letting Q = H/e Fe, we get that Q is 2-crystallographic, as it inherits an effective cocompact action

2 on E .

By the above work, we have therefore proven the following result:

Proposition 3.16. Let M be a compact Seifert ﬁber space with nonempty boundary, and let

Γ = π1(M) be the fundamental group. Let B be the hyperbolic base orbifold with virtually free

orb orbifold fundamental group π1 (B) = Γ0. Let T be the Bass-Serre covering tree of the graph

of groups (Γ0,Y ) given by Theorem 3.11. Let φ :Γ → Γ0 be the quotient map, and let A be

63 the collection of preimages of maximal inﬁnite virtually cyclic subgroups of Γ0. Let H be a set of representatives of conjugacy classes in A. Consider the following cellular Γ-pushout:

a Γ × T He E He∈H

a Γ × EH X. He e He∈H

Then X is a model for EΓ. In the above cellular Γ-pushout, we require either (1) α is the

disjoint union of cellular He-maps (He ∈ H), β is an inclusion of Γ-CW-complexes, or (2) α is the disjoint union of inclusions of He-CW-complexes (He ∈ H), β is a cellular Γ-map.

Corollary 3.17. Let M be a compact Seifert ﬁber space with nonempty boundary, and let

Γ = π1(M) be the fundamental group. Then gd(Γ) = 3.

Proof. The model constructed is three dimensional, as the EHe are three dimensional (The-

2 orem 1.50); this implies gd(Γ) ≤ 3. But Γ has a subgroup isomorphic to Z (take any 2 subgroup in A), so by Corollary 1.4 gd(Γ) ≥ gd(Z ) = 3. Thus, gd(Γ) = 3.

2 3.1.2 Closed base orbifolds modeled on H

Let M be a closed Seifert ﬁber space with fundamental group π1(M) = Γ, and suppose that

2 the base orbifold B is modeled on H . Let Γ0 = π1(B) be the fundamental group of B; 2 ∼ then Γ0 ≤ Isom(H ) = PSL2(R) o Z2, and by Lemma 3.2 we have the following short exact sequence:

1 → Z → Γ → Γ0 → 1.

By Theorem 3.3, B is ﬁnitely covered by a manifold Be. Since Be is then a closed, compact

2 2 quotient manifold of H , π1(Be) is a lattice in Isom(H ); as Γ0 is a finite extension, it is also 2 a lattice in Isom(H ). 64 Before we describe A and Ae, we first define and characterize certain subgroups of Γ0.

2 For any ξ ∈ ∂H , let Pξ ≤ Γ be the subgroup of nonhyperbolic isometries of Γ that ﬁx ξ; 2 without loss of generality assume that point is ∞ (within Isom(H ), one can conjugate Γ0 by an appropriate rotation moving ξ to ∞). Then the orientation-preserving component

2 of P∞ must be a discrete subgroup of parabolic isometries, as rotations act freely on ∂H . 2 ◦ 2 ∼ Consider the upper half-plane model for H ; then Isom (H ) = PSL2(R) in the obvious way, 1 α and orientation-preserving isometries in P are simply matrices of the form , or ∞ 0 1 2 2 horizontal translations of H . P∞ thus acts discretely on any horizontal line in H , so that ◦ P∞ is a discrete subgroup of Isom (E). P∞ is therefore congruent to one of 1, Z2, Z or ZoZ2;

in particular, Pξ is a virtually cyclic subgroup, generated by at most a parabolic isometry

that ﬁxes ξ and a reﬂection through a geodesic c(R) with c(∞) = ξ.

This idea generalizes to higher-dimensional hyperbolic geometries as well; a subgroup P∞

n of isometries that fix ∞ ∈ ∂H also fix horizontal hyperplanes in the upper half-space model n for H ; these hyperplanes are a specific instance of the more general concept of a horosphere (see [BH99, Chapter II.8]).

2 Proposition 3.18. Let Γ0 be a lattice in Isom(H ), and let FIN 0 and VC0 be the families

of ﬁnite and virtually cyclic subgroups in Γ0, respectively. Then the collection A of maximal

inﬁnite virtually cyclic subgroups of Γ0 is adapted to the pair (FIN 0, VC0).

Proof. First, we discuss the types of elements that lie in A. Since A ∈ A must be inﬁnite

2 virtually cyclic, by Proposition 1.27 either A stabilizes a unique geodesic c(R) ⊂ H or ﬁxes 2 a unique boundary point ξ ∈ ∂H , depending on whether A acts semi-simply. If it does we

denote the subgroup Mc; if it does not we denote it Pξ.

We now show that A is adapted to the pair (FIN 0, VC0) by checking the conditions directly.

1. Suppose Mc,Pξ ∈ A for some geodesic c and boundary point ξ. Any orientation-

preserving nontrivial isometry is either parabolic and therefore not in Mc or not 65 parabolic and therefore not in Pξ; thus Mc ∩ Pξ consists only of elliptic elements.

2 Since rotations act freely on ∂H they cannot ﬁx ξ, so Mc ∩ Pξ must consist only

of reflections that fix ξ. Suppose ρ, σ ∈ Mc ∩ Pξ are distinct reflections; then ρσ is a nontrivial orientation-preserving isometry that fixes ξ. ρσ is therefore not elliptic,

contradicting the fact that Mc ∩ Pξ has no infinite order elements. Mc ∩ Pξ is therefore generated by no more than one reflection, and in particular is finite.

0 Suppose Mc,Mc0 ∈ A are distinct; then c 6= c . By Corollary 1.24, hyperbolic elements

stabilize unique geodesics, so M = Mc ∩ Mc0 contains only reﬂections and rotations. Consider the orientation-preserving subgroup M ◦ of M; it is a group of rotations that

stabilize both c and c0. A rotation that stabilizes a geodesic has a ﬁxed point on that

geodesic; thus, either M ◦ is trivial (and in particular ﬁnite) or c and c0 intersect at

the ﬁxed points of the rotations in M ◦. Suppose distinct rotations in M ◦ have ﬁxed

points x and y; then by Corollary 1.10, as there is a unique geodesic line connecting

2 0 any two distinct points in H and since c 6= c by supposition, we must have y = x. ◦ 2 Thus, M ≤ (Γ0)x for some x ∈ H , which is finite by Lemma 1.17. We therefore have ◦ that M has an index at most 2 subgroup M which is finite, so that M = Mc ∩ Mc0 is finite.

0 Suppose Pξ,Pξ0 ∈ A, ξ 6= ξ . Then an element γ ∈ Pξ ∩ Pξ0 must stabilize the unique

0 geodesic c(R) satisfying c(∞) = ξ, c(−∞) = ξ (Corollary 1.23). Since γ ∈ Pξ cannot 2 be hyperbolic, it must be elliptic; the only elliptic isometries of H that ﬁx the points

c(∞) and c(−∞) are the reﬂection through c and the identity. Thus, Pξ ∩ Pξ0 is either

trivial or isomorphic to Z2, and is therefore ﬁnite.

−1 2 2. Let Mc ∈ A, γ ∈ Γ0. Then γMcγ = Mγ.c ∈ A. Similarly, for any ξ ∈ ∂H and −1 γ ∈ Γ0, γPξγ = Pγ.ξ ∈ A.

−1 3. Let Mc ∈ A for some geodesic c(R), and let n ∈ NΓ0 (Mc). Then nMcn ≤ Mc

66 2 stabilizes both n.c(R) and c(R). H is CAT(-1), so by Proposition 1.27, n.c(R) = c(R).

Then n ∈ Mc and thus NΓ0 (Mc) = Mc.

2 −1 Let Pξ ∈ A for some ξ ∈ ∂H and let n ∈ NΓ0 (Pξ); then nPξn ≤ Pξ ﬁxes n.ξ and ξ. −1 Since A is conjugacy closed, nPξn = Pn.ξ ∈ A; since Pnξ ≤ Pξ and Pn.ξ is maximally

infinite virtually cyclic, we have that Pn.ξ = Pξ. Let p ∈ Pξ be a generator of an infinite finite-index subgroup; then p is parabolic and fixes both n.ξ and ξ. By Corollary 1.26

it must be that n.ξ = ξ, so n ﬁxes ξ.

Suppose n is hyperbolic; then n stabilizes a unique geodesic c(R) satisfying c(∞) = ξ. By conjugating so that ξ corresponds to the equivalence class of vertical lines in the

2 upper half-plane model for H and so that c is the positive imaginary axis, we may 1 α β 0 assume p = and n = for some α > 0, β > 1. But then for any 0 1 0 β−1 k ∈ Z, we have

−2k 1 αβ k→∞ n−kpnk = → I. 0 1

This set of elements belonging to Γ0 contradicts the assumption that Γ0 acts discretely.

Thus n cannot be hyperbolic, so n is either a parabolic or elliptic element of Γ0 that

ﬁxes ξ. This forces n ∈ Pξ, and so NΓ0 (Pξ) = Pξ.

4. Let H be an inﬁnite virtually cyclic subgroup of Γ0, and suppose ﬁrst that H acts

semi-simply, stabilizing a unique geodesic c. Let Mc ≤ Γ0 be the (nonempty) stabilizer

of c(R); then by Proposition 1.28 Mc is virtually cyclic. Let m ∈ Mc generate a ﬁnite

index subgroup of Mc, and assume N is a VC subgroup with Mc ≤ N. Then N acts

2 semi-simply on H , since if N had a parabolic element n then hni ∩ hmi 6= {1}; this implies some nontrivial power of n is both parabolic and hyperbolic, a contradiction.

But then N stabilizes the same geodesic line that Mc does, so that N ≤ Mc. Mc is

therefore maximal, and thus H ≤ Mc ∈ A.

67 Now let H contain a parabolic element; then H ﬁxes a unique boundary point ξ.

Then H ≤ Pξ, since H cannot contain any hyperbolic isometries (due to the above

2 argument) or rotations (which act freely on ∂H ). Suppose some virtually cyclic N

contains Pξ; then N contains a parabolic element that ﬁxes ξ, so immediately N ≤ Pξ.

Pξ is therefore maximal, and H ≤ Pξ ∈ A.

With the adapted collection sorted, we construct the models needed for the pushout

construction.

2 2 Claim 3.19. Let Γ0 be a lattice in Isom(H ). Then H is a model for EΓ0.

2 H Proof. Let H be a ﬁnite subgroup of Γ0; then by Corollary 1.19 (H ) is contractible.

If H is an infinite subgroup of Γ0, then either H has only infinitely many rotations or an infinite element (the former case is not actually possible, but it is easier to show that it is

harmless than irrelevant). Since Γ0 is discrete it cannot have inﬁnitely many rotations about a single point. If Γ has rotations about distinct points then no single point is ﬁxed by all of

◦ 2 2 Γ. Any inﬁnite element of a discrete subgroup of Isom (H ) has no ﬁxed points in H . In 2 H 2 any case it must be that (H ) is empty, so H is shown to be a model for EΓ0.

In order to construct models pertaining to the subgroups He ∈ H, we demonstrate ﬁrst that they are virtually 2-crystallographic.

φ Lemma 3.20. Let 0 → Z → Ge → G → 0 be a short exact sequence of groups and let H be an inﬁnite virtually cyclic subgroup of G. Then there exists a normal subgroup of ﬁnite −1 ∼ 2 ∼ index Ke E He = φ (H) with either Ke = Z or Ke = Z o Z.

Proof. Let K be the normal inﬁnite cyclic subgroup of H given by Lemma 1.2, and let

−1 −1 −1 Ke = φ (K). Then for any h ∈ He, φ(h Khe ) = φ(h) Kφ(h) ≤ K, so Ke E He.

68 ∼ ∼ Let a ∈ Ke be a generator of the kernel ker φ = Z, and let b ∈ Ke satisfy K = hφ(b)i. ∼ Then we know hai ∩ hbi = {0}, and that hai = ker φ E Ke. So Ke = hai oϕ hbi, where ∼ ϕ ∈ Aut(Z) = {±1}.

Proposition 3.21. All subgroups in the collection Ae have a ﬁnite index subgroup that is 2-crystallographic.

Proof. Let Pξ ∈ A be orientation-preserving; then Pξ contains only parabolic elements, so ∼ Pξ = Z. By Lemma 3.2 we have the exact sequence 1 → Z → Pfξ → Pξ → 1. By the argument 2 ∼ 2 in the proof of Lemma 3.20, Pfξ is isomorphic either to Z or to Z o Z = Z o Z2. The former case is trivially 2-crystallographic. In the latter case, since the action of Z2 is eﬀective the 2 normal Z subgroup is maximal abelian; by Theorem 1.45 Pfξ is again 2-crystallographic.

Now suppose Mc ∈ A is orientation-preserving; then Mc consists of hyperbolic translations along the geodesic line c(R) and rotations about some point on c(R). Rotations about ∼ ∼ two distinct points compose to a translation, so either Mc = hγi = Z or Mc = hγi o hσi =

ZoZ2 for some translation γ and some rotation σ. The former case is algebraically equivalent ∼ to the Pξ case. Assume Mc = ZoZ2; by the construction of Ke in Lemma 3.20, Mfc = Ke ohσei for some σe satisfying φ(σe) = σ. As σ acts nontrivially by conjugation on γ in Γ0, σe must act nontrivially on any γe ∈ Ke with φ(γe) = γ. Thus, Mfc/Ke acts eﬀectively on Ke, and we again have that Mfc is 2-crystallographic. Finally, suppose that A ∈ A is not orientation-preserving; then it has a subgroup A◦ of index 2 that is. The preimage Af◦ is 2-crystallographic by the previous cases and is an index

2 subgroup of Ae by the Lattice Isomorphism Theorem.

As models for classifying spaces with ﬁnite or virtually cyclic isotropy are known for virtually crystallographic groups, we may now apply the pushout construction.

2 Proposition 3.22. Let M be a closed Seifert ﬁber space with base orbifold B modeled on H .

Let Γ = π1(M) and Γ0 = π1(B) be the respective fundamental groups. Let A be the collection 69 of maximal inﬁnite virtually cyclic subgroups of Γ0, let Ae be the collection of preimages of

A in Γ, and let H be a set of representatives of conjugacy classes in Ae; then all He ∈ H

are virtually 2-crystallographic. Let EHe be the models constructed in Proposition 1.49, and consider the following cellular Γ-pushout:

a Γ × 2 He E H He∈H

a Γ × EH X. He e He∈H

Then X is a model for EΓ. In the above cellular Γ-pushout, we require either (1) α is the

disjoint union of cellular He-maps (He ∈ H), β is an inclusion of Γ-CW-complexes, or (2) α is the disjoint union of inclusions of He-CW-complexes (He ∈ H), β is a cellular Γ-map.

Proof. By Proposition 3.18, A is adapted to the pair (FIN 0, VC0) of families of subgroups

2 in Γ0. This allows us to apply the construction of Corollary 2.5. By Claim 3.19, H is a model for EΓ0. Let H ∈ A. By Corollary 1.24, H is the stabilizer of a unique geodesic c in

2 H . The subgroup of H that ﬁxes c is ﬁnite and normal; the quotient of H by this group

inherits an eﬀective action on c ' E, and is therefore 1-crystallographic. This gives E as a model for EH by Proposition 1.3. By Proposition 3.21 all subgroups He ∈ H are virtually

2-crystallographic; in particular, by the argument in Claim 3.15, He is a ﬁnite extension of a 2-crystallographic group Q. Again by Proposition 1.3 models with ﬁnite or virtually cyclic isotropy can be constructed for Q and given He-actions. By Proposition 1.49 we have models for EQ that are thus models for EHe as well.

2 Corollary 3.23. Let M be a closed Seifert ﬁber space with base orbifold modeled on H , and let Γ = π1(M) be the fundamental group of M. Then gd(Γ) = 3.

2 Proof. The model constructed is three dimensional, as EHe and E × [0, 1] are both three 2 dimensional; this gives that gd(Γ) ≤ 3. Γ has a subgroup isomorphic to Z (consider any 70 2 2 He ∈ H), so by Corollary 1.4 we know that gd(Z ) ≤ gd(Γ). By Theorem 1.50 gd(Z ) = 3, so it must be that gd(Γ) = 3.

2 Corollary 3.24. Let Γ be a lattice in PSLg 2(R) or Isom(H × E); then gd(Γ) = 3.

Proof. Let Γ be a uniform lattice; then the closed manifold M created by taking the quotient

2 of the appropriate geometry (PSLg 2(R) or H × E) by Γ is a Seifert ﬁber space with base 2 orbifold modeled on H by Theorem 3.7. The result then follows from Corollary 3.23. If Γ is a nonuniform lattice, M is not closed; it is ﬁnite volume, however, so the manifold ¯ ¯ M = M ∪ ∂M is compact with fundamental group π1(M) = Γ. As Γ is not virtually cyclic

2 (it contains subgroups isomorphic to Z ), the result follows from Corollary 3.17.

2 3.1.3 Closed base orbifolds modeled on E

2 If M is a closed Seifert ﬁber space with base orbifold B modeled on E , then we know by 3 Theorem 3.7 that M is modeled on either E or Nil. Unfortunately there is not an obvious choice for the adapted collection A; in particular, the collection of maximal inﬁnite stabilizers

2 of geodesics in E fails to satisfy the self-normalizing condition in general. Fortunately, if Γ is the fundamental group of such a manifold, gd(Γ) has already been

computed in the literature. For completeness, we summarize that work here.

3 3 Suppose ﬁrst that M is modeled on E ; then Γ is a uniform lattice in Isom(E ), and is therefore by deﬁnition 3-crystallographic; we remind the reader that Connolly, Fehrman,

and Hartglass demonstrated that gd(Γ) = 4 and constructed a model of minimal dimension

(see Proposition 1.49 and Theorem 1.50, or [CFH08]).

When M is modeled on Nil, L¨uck and Weiermann have demonstrated ([LW12]) that gd(Γ) = 3; however, the model their method creates is of dimension 4, and no minimal

dimension model is known to the author. What follows is a demonstration of their work and

how it applies to uniform lattices in Isom(Nil).

71 Let FIN and VC be the families of ﬁnite and virtually cyclic subgroups of a group Γ, respectively. Deﬁne an equivalence relation on VC\FIN in the following way:

V,W ∈ VC \ FIN satisfy V ∼ W if |V ∩ W | = ∞.

Note that for V,W ∈ VC \ FIN and γ ∈ Γ, V ⊆ W ⇒ V ∼ W and V ∼ W ⇒ γ−1V γ ∼

γ−1W γ. Let [VC\FIN ] be the set of equivalence classes of ∼, with [V ] ∈ [VC\FIN ] the equivalence class of V ∈ VC \ FIN . Deﬁne the subgroup

−1 NΓ[V ] := {γ ∈ Γ | [γ V γ] = [V ]}.

This subgroup is this isotropy group of [V ] under the Γ-action on [VC\FIN ] induced by

conjugation. Let FIN |NΓ[V ] be the family of ﬁnite subgroups of NΓ[V ], and let

VC[V ] := {W ⊆ NΓ[V ] | W ∈ VC \ FIN , [W ] = [V ]} ∪ FIN |NΓ[V ].

It is clear that this set is closed under conjugation and passing to subgroups, and so is a family of subgroups of NΓ[V ].

Deﬁnition. A group Γ is poly-Z if there is a ﬁnite sequence

{1} = G0 ≤ G1 ≤ ... ≤ Gn = Γ

∼ of subgroups such that Gi−1 E Gi with Gi/Gi−1 = Z for all i ∈ {1, 2, . . . , n}. A group is virtually poly-Z if it has a ﬁnite-index subgroup that is poly-Z.

We can now present the result of L¨uck and Weiermann ([LW12, Theorem 5.13]).

Theorem 3.25. Let Γ be a virtually poly-Z group which is not virtually cyclic. Let [VC\

FIN ]f be the subset of [VC\FIN ] consisting of those elements [V ] for which NΓ[V ] has ﬁnite index in Γ. Let [VC\FIN ]f /Γ be its quotient under the Γ-action coming from conjugation. Then

gd(Γ) = vcd(Γ), and precisely one of the following occurs: 72 1. The set [VC\FIN ]f /Γ is empty. This is equivalent to the condition that every ﬁnite index subgroup of Γ has a ﬁnite center. In this case,

gd(Γ) = vcd(Γ);

2. The set [VC\FIN ]f /Γ contains exactly one element [V ] · Γ. this is equivalent to the condition that there exists an inﬁnite normal cyclic subgroup C ⊆ Γ and for every

inﬁnite cyclic subgroup D ⊆ Γ with [Γ : NΓD] < ∞ we have C ∩ D 6= {1}. Then there is the following dichotomy:

(a) For every [W ]·Γ ∈ [VC\FIN ]/Γ such that [W ]·Γ 6= [V ]·Γ we have vcd(NΓ[W ]) ≤ vcd(Γ) − 2. In this case, vcd(Γ) ≥ 4 and

gd(Γ) = vcd(Γ) − 1;

(b) There exists [W ]·Γ ∈ [VC\FIN ]/Γ such that [W ]·Γ 6= [V ]·Γ and vcd(NΓ[W ]) = vcd(Γ) − 1. In this case, vcd(Γ) ≥ 3 and

gd(Γ) = vcd(Γ);

3. The set [VC\FIN ]f /Γ contains more than one element. This is equivalent to the condition that there is a ﬁnite-index subgroup of Γ whose center contains a subgroup

2 isomorphic to Z . In this case,

gd(Γ) = vcd(Γ) + 1.

We would like to demonstrate that if Γ is a uniform lattice in either Isom(Nil) or Isom(Sol) then Γ is virtually poly-Z, as well as compute the dimension vcd(Γ). Again, L¨uck and Weiermann have a related result ([LW12, Lemma 5.14,i-iv]).

Lemma 3.26. 1. Subgroups and quotients of poly-Z groups are again poly-Z. If 0 →

G0 → G1 → G2 → 0 is an extension of groups and G0 and G2 are poly-Z groups, then

G1 is a poly-Z group. 73 The same statements are true if one replaces “poly-Z” by “virtually poly-Z” everywhere;

d 2. Let G be a poly-Z group of cohomological dimension cd(G) = d. Then H (G) is

isomorphic to Z or Z/2Z;

3. Let G be a poly-Z group. Suppose there is a ﬁnite sequence

{1} = G0 ≤ G1 ≤ ... ≤ Gn = G

of subgroups such that Gi−1 is normal in Gi with inﬁnite cyclic quotient Gi/Gi−1 for i = 1, 2, . . . , n. Then

n = cd(G);

4. If 0 → G0 → G1 → G2 → 0 is an extension of poly-Z groups, then

vcd(G1) = vcd(G0) + vcd(G2).

We wish to demonstrate that Γ = π1(M) ≤ Isom(Nil) is virtually poly-Z.

Proposition 3.27. Let M be a closed Seifert ﬁber space modeled on Nil with base orbifold

2 B modeled on E . Let Γ = π1(M) be the fundamental group of M; then Γ is virtually poly-Z with vcd(Γ) = 3.

Proof. Let Γ0 = π1(B) be the fundamental group of the base orbifold. By Lemma 3.2, we have the exact sequence 1 → Z → Γ → Γ0 → 1. Since B is a closed orbifold modeled 2 2 on E ,Γ0 is a uniform lattice in Isom(E ), and is therefore a 2-crystallographic group; by 2 Proposition 1.44, Γ0 is virtually Z , and is therefore virtually poly-Z with vcd(Γ0) = 2. By

Lemma 3.26, since Γ is an extension of the poly-Z group Γ0 by Z, Γ is itself poly-Z with vcd(Γ) = 3.

74 In order to place Γ in the proper case of Theorem 3.25, we need to understand Nil on a more elemental level. We will consider Nil to be homeomorphic to the continuous Heisenberg

Lie group; that is,  1 x z   ∼   Nil = H3(R) =  0 1 y  : x, y, z ∈ R ,  0 0 1  with product given by matrix mutiplication. Alternatively, Nil can be identiﬁed with the

3 group R with multiplication given by (a, b, c)·(d, e, f) = (a+d, b+e, c+f −ae). The center of Nil is the subgroup {(0, 0, z): z ∈ R}. It is a simple matter to compute conjugates and positive powers:

(x, y, z)−1 · (a, b, c) · (x, y, z) = (a, b, ay − bx + c);

n(n − 1) (a, b, c)n = na, nb, nc − ab for n > 0. 2

It is known (see [Sco83, p. 467] or [Thu97, Sections 3.8, 4.7]) that isometries of Nil are orientation-preserving and lie in a short exact sequence of the form

p 2 0 → E → Isom(Nil) → Isom(E ) → 0, and that Isom(Nil) has two components (the identity component and the other consisting

2 of isometries that ﬂip the orientation of the base space E and the vertical component E). Let Isom◦(Nil) denote the identity component of Isom(Nil); then Isom◦(Nil) ∼= Nil, where elements of the Lie group Nil act on themselves by matrix multiplication on the left.

Proposition 3.28. Let M be a closed Seifert ﬁber space modeled on Nil, and let Γ = π1(M) be its fundamental group. Then gd(Γ) = 3.

Proof. Assume without loss of generality that Γ is contained in the identity component of Isom(Nil); if it is not, we can use Proposition 1.3 to get that gd(Γ) = gd(Γ0), where

Γ0 = Γ ∩ Isom◦(Nil) is an index 2 subgroup of Γ.

75 We endeavor to show that Γ satisﬁes the conditions of Case 2b in Theorem 3.25. The

inﬁnite normal cyclic subgroup C ⊆ Γ is the center Z(Γ) of Γ.

If (a, b, c) ∈ Nil generates an inﬁnite cyclic subgroup D of Γ, it is a simple calculation to

show that for any (x, y, z) ∈ Nil,

(x, y, z)−1(a, b, c)(x, y, z) = (a, b, ay − bx + c).

So if (x, y, z) ∈ NΓD then either n = 1 and ay − bx = 0 simultaneously, or a = b = 0. If a = b = 0, then D ≤ Z(Γ); in particular, Z(Γ) ∩ D 6= {1}.

Suppose n = 1, and without loss of generality suppose a 6= 0. Then for (x, y, z) to be a b normalizer of D, we need ay − bx = 0, or y = x. Thus, a b NΓD = α, α, β α, β ∈ R ∩ Γ. a

Our claim is that [Γ : NΓD] = ∞, so that Γ satisﬁes the conditions for Case 2. Note ﬁrst

that the group N generated by D and Z(Γ) satisﬁes N ≤ NΓD, and since Z(Γ) is central

we have that N is an extension of D by Z(Γ); in particular, N is poly-Z with vcd(N) = 2. bα The elements in N D with α = 0 belong to Z(Γ); let η = α, , β ∈ N D be such Γ a Γ that α 6= 0. Then powers of η also belong to NΓD, as do powers of (a, b, c). For m, n > 0, consider elements of the form nbα a−mηn = nα − ma, − mb, ∗ ; a If α is not a rational multiple of a, we can choose positive m and n such that nα − ma and nbα b −mb = (nα−ma) are simultaneously arbitrarily close to elements in the center of Nil. a a

By considering the images of these elements in Γ0 = π1(B), we can construct a sequence of

isometries in Γ0 that is indiscrete. But Γ0 must be discrete, as B is a closed orbifold; this is

a contradiction. Thus, α is a rational multiple of a, which means all elements in NΓD \ Z(Γ)

k have a ﬁnite power that can be presented as (a, b, c) · z for some k ∈ Z and some z ∈ Z(Γ). Since Γ is also discrete, there must be some maximal such power (or alternatively, some minimal α > 0). Thus [NΓD : N] < ∞, so NΓD is virtually poly-Z with vcd(NΓD) = 2. 76 Since vcd(Γ) = 3, it must be that [Γ : NΓD] = ∞. Thus, for any inﬁnite cyclic D ≤ Γ, either

Z(Γ) ∩ D 6= {1} or [Γ : NΓD] = ∞; in particular, Γ satisﬁes the conditions of Case 2. Finally, since vcd(Γ) = 3, we cannot be in Case 2a. So Γ must satisfy the conditions of

Case 2b of Theorem 3.25, and therefore gd(Γ) = vcd(Γ) = 3.

3 3.2 Compact Manifolds modeled on Sol or H

Let M be a compact manifold modeled on Sol; then Γ = π1(M) is a uniform lattice in Isom(Sol). The following result of Thurston ([Thu97, Theorem 4.7.13]) lets us immediately

show that Γ is poly-Z:

Theorem 3.29. A group Γ is a cocompact discrete group of automorphisms of solvgeometry

if and only if Γ contains a subgroup H of ﬁnite index whose centralizer is trivial and that is

an extension of the form

2 Z → H → Z,

2 where the action of Z on Z is generated by a hyperbolic element of SL2(Z).

Corollary 3.30. Let Γ ≤ Isom(Sol) be a uniform lattice; then Γ is poly-Z with vcd(Γ) = 3.

Proof. By Theorem 3.29, Γ has a ﬁnite-index subgroup that is an extension of poly-Z groups. Lemma 3.26 gives the result.

We’d like to ﬁnd the set [VC/FIN ]f for Γ. Again we look to [Sco83, p. 470] and [Thu97, Sections 3.8, 4.7] for general knowledge of Sol. In particular, Sol is known to sit in its

2 isometry group as a subgroup of ﬁnite index 8. It is an extension of R by R ,

2 p 0 → R → Sol → R → 0,

where the action of Sol on itself is given by

−t t (x, y, t)(a, b, s) = (x + e a, y + e b, t + s), a, b, s, t, x, y ∈ R. 77 Note that ker p = {(x, y, 0) ∈ Sol | x, y ∈ R}. Then conjugating (x, y, t) by (a, b, s) gives

(a, b, s)−1(x, y, t)(a, b, s) = (esx + esa(e−t − 1), e−sy + e−sb(et − 1), t).

Suppose C = h(x, y, t)i is an inﬁnite cyclic subgroup of Sol. Note that

(esx + esa(e−t − 1), e−sy + e−sb(et − 1), t) = (x, y, t)−1 = (−xet, −ye−t, −t)

forces t = 0; reducing the equation gives

(esx, e−sy, 0) = (−x, −y, 0),

which requires x = y = 0. Thus, NSol(C) = CSol(C) for any inﬁnite cyclic C ≤ Sol.

s −s Note ﬁrst that if C = h(x, y, 0)i, then (a, b, s) ∈ CSol(C) implies (e x, e y, 0) = (x, y, 0), which is true if and only if s = 0 or x = y = 0. Thus, C ≤ ker p an inﬁnite cyclic subgroup

implies that CSol(C) = ker p.

If now C = h(x, y, t)i with t 6= 0, then (a, b, s) ∈ CSol(C) implies

x(e−s − 1) = a(e−t − 1), y(es − 1) = b(et − 1).

Suppose s = αt for some α ∈ R; then we can write e−αt − 1 eαt − 1 a = x, b = y, e−t − 1 et − 1

so that (a, b, s) can be described entirely by the generator (x, y, t) of C and a choice of general

α ∈ R. Moreover, for α, β ∈ R, e−αt − 1 eαt − 1 e−βt − 1 eβt − 1 x, y, αt · x, y, βt e−t − 1 et − 1 e−t − 1 et − 1 e−(α+β)t − 1 e(α+β)t − 1 = x, y, (α + β)t . e−t − 1 et − 1 e−αt − 1 eαt − 1 So N (C) = C (C) ∼= via x, y, αt 7→ α, with C corresponding to the Sol Sol R e−t − 1 et − 1 subgroup Z ≤ R.

Proposition 3.31. Let Γ be a uniform lattice in Isom(Sol); then gd(Γ) = 3. 78 0 Proof. By Corollary 3.30, Γ is virtually poly-Z; let Γ = Γ ∩ Sol. Let C ∈ VC \ FIN . Then 0 0 NΓ(C) ≤ NΓ0 (C ∩ Γ ), so [Γ : NΓ(C)] ≥ [Γ : CΓ0 (C ∩ Γ )] = ∞; thus, [VC\FIN ]f = ∅ and so Γ satiﬁes the conditions for Theorem 3.25 Case 1. As vcd(Γ0) = 3, gd(Γ) = vcd(Γ0) = 3.

Note that the proof of Theorem 3.25 in Luck and Weiermann ([LW12, Theorem 5.13])

gives a model for EΓ of minimal dimension: Let I be a collection of inﬁnite cyclic subgroups

C ≤ Γ such that only one representative of each η ∈ [VC\FIN ]/Γ is chosen satisfying

NΓ[C] = NΓ(C). Then the Γ-pushout

a Γ ×NΓ(C) ENΓ(C) EΓ C∈I

a Γ ×NΓ(C) EWΓ(C) X C∈I

yields a Γ-CW-complex X that is a model for EΓ, where WΓ(C) is the Weyl group deﬁned

by WΓ(C) = NΓ(C)/C. In particular, we have shown that NΓ(C) = CΓ(C) for all inﬁnite cyclic C.

◦ To ﬁnd models for ECΓ(C) and EWΓ(C), we note that if Γ is the identity component of ∼ ◦ ∼ 2 Γ, then CΓ◦ (C) was shown to be C = Z (resp. Γ ∩ ker p = Z ) if C = h(x, y, t)i with t 6= 0 ◦ (resp. t = 0). As [Γ : Γ ] ≤ 8, we thus have that CΓC is isomorphic to a ﬁnite extension of

2 either Z or Z , while WΓ(C) is isomorphic to either a ﬁnite subgroup or a ﬁnite extension of

Z. By the usual trick of quotienting out by a ﬁnite subgroup acting trivially on the normal n free abelian subgroup and applying Proposition 1.3, we thus have E as a model for ECΓ(C)

or EWΓ(C), where n is the rank of the free abelian subgroup of CΓ(C) or WΓ(C). Note that for Γ ≤ Isom(Nil), the equivalent model is four-dimensional. L¨uck and Weier-

mann have a Bredon cohomology-based argument for why this is too large in that case, so

we know that there is a model for EΓ of dimension precisely three; the explicit construction of such a model remains an open problem.

79 3 Now let M be a ﬁnite-volume manifold modeled on H , and let Γ = π1(M) be the 3 fundamental group. As H is itself a space of constant negative curvature, we can skip the projection and lifting of adapted collections and use Lafont and Ortiz’s pushout model

directly ([LO09a, Proposition 3.1]).

3 Let us ﬁrst discuss a way of thinking of the elements in Isom(H ). This discussion comes 3 3 largely from [Mas88] and [Sco83]. Let H be the upper half 3-space R+ = {(x, y, z) | z > 0}. 3 3 Isometries of H are in fact determined by their action on ∂H ; thinking of the boundary as 2 the one point compactiﬁcation of the z = 0 plane Cˆ = C∪{∞} ' S , we can therefore think of orientation-preserving isometries as M obius transformations of the boundary. In fact, if

3 we identify (x, y, z) ∈ H with the quaternion w = x+yi+zj, then we can extend the action a b aw + b of a matrix α = ∈ PSL ( ) to an action on 3 in the obvious way: w 7→ . c d 2 C H cw + d 3 Moreover all orientation-preserving isometries of H can be obtained in this way.

Let us ﬁrst characterize the matrices in PSL2(C) that correspond to elliptic, hyperbolic and parabolic isometries.

Proposition 3.32 ([Mas88, Section I.B]). Let α ∈ PSL2(C) be a nontrivial, orientation pre- 3 2 serving isometry of H ; then Tr α—the square of the trace of α—determines the isomorphism type in the following way:

• If Tr2α = 4, then α is parabolic;

• If Tr2α ∈ [0, 4), then α is elliptic;

2 • If Tr α ∈ C is any other value, then α is hyperbolic.

Proof. As there are no free actions on S2 and an isometry that ﬁxes two boundary points

must also ﬁx the unique geodesic that connects them by Corollary 1.23, α ﬁxes either 1 or

3 2 points in ∂H . Suppose first that α has only one fixed boundary point; by conjugation (recall that trace is fixed under conjugation), we may assume that α fixes ∞. Then α must

80 1 λ correspond to a free action on , and is therefore a translation. We can write α = C 0 1 2 for some nonzero λ ∈ C, and Tr α = 4. Suppose now that α has two ﬁxed points; by conjugation, we assume they are 0 and ∞,

3 and that α therefore ﬁxes the vertical z-axis in the upper half-space model of H . α then iθ must have the form α(w) = λw for some λ ∈ C with λ 6= 0, 1. Let λ = re , r ∈ R positive r1/2eiθ/2 0 and θ ∈ [0, 2π); we may write α = . If |λ| = 1, then α is simply a 0 r1/2eiθ/2 rotation about the z-axis by the angle θ, and α is elliptic; Tr2α = 2 + 2 cos θ ∈ [0, 4) (if

|λ| = 1 and θ = 0 then λ = 1, which we avoid). If |λ|= 6 1, then α has nontrivial action on 1 the vertical z-axis, so α is hyperbolic. If λ is real then θ ∈ {0, π}, so Tr2α = 2± |λ| + , |λ| 2 2 and therefore Tr α ∈ R \ [0, 4]; if λ is not real, then Tr α is not real.

Remark 3.33. The hyperbolic isometries correspond to screw actions along their stabilized geodesics, or combinations of a nontrivial translation with a (possibly trivial) rotation about the axis. Some authors refer only to irrotational hyperbolic isometries as hyperbolic, calling rotational translations loxodromic. Some authors call the general case loxodromic. In order to remain consistant, we will call all such isometries hyperbolic.

3 We know that Γ is a lattice in Isom(H ); we wish to make A the collection of maximally 3 3 infinite subgroups that either stabilize a geodesic c(R) ⊂ H or fix a boundary point ξ ∈ ∂H . 3 As these subgroups are more complicated in the H setting, we discuss them first. See [Sco83, 3 p. 448] for details about the geometry of H . We first note that hyperbolic and parabolic isometries can’t live together in such a sub-

group discretely.

3 Proposition 3.34. Let P be a group of isometries that ﬁx a boundary point ξ ∈ ∂H and that contains hyperbolic and parabolic elements. Then P is not discrete.

Proof. Let h and p be hyperbolic and parabolic isometries in P , respectively; assume further

that h and p are orientation-preserving (by squaring if necessary). Then h must stabilize 81 3 some geodesic c(R) ⊂ H satisfying c(∞) = ξ; by conjugating properly, assume ξ is the point 3 at inﬁnity and c is the vertical z-axis in the upper half-space model for H . Then h is a screw motion about the vertical axis and p is a horizontal translation by some vector v = ha, b, 0i.

By switching to h−1 if necessary, we identify β 0 1 α h = , p = 0 β−1 0 1 1 αβ−2n for some β ∈ , α ∈ with |β| > 1, α 6= 0. Then h−nphn = , so R C 0 1 1 0 h−nphn → as n → ∞. This sequence clearly violates discreteness in the compact- 0 1 open topology.

3 3 Claim 3.35. Let Γ ≤ Isom(H ) be a lattice, and let ξ ∈ ∂H . Then the group P of parabolic 2 elements that ﬁx ξ is either trivial or isomorphic to Z .

Proof. Since Γ is a lattice, we should be able to construct a convex fundamental domain for

3 H /Γ with a cusp at ξ; by conjugation, we assume ξ = ∞. 3 Consider any horizontal plane in the half-space model for H ; the parabolic elements that ﬁx ∞ will be horizontal translations, and will therefore ﬁx this plane. Thus, we may consider

2 2 P as a group of translations acting on E , where E is represented by any horizontal plane that intersects the chosen fundamental domain with cusp at ∞. But since the fundamental

domain has ﬁnite volume deﬁned as an integral of the surface area of cross-sections, we

2 must have that this intersection has ﬁnite area. So P must be a lattice in Isom(E ), and is 2 therefore isomorphic to Z .

3 Claim 3.36. Let Γ ≤ Isom(H ) be a lattice, and deﬁne a subgroup Pξ = FixΓ(ξ) to be the 3 elements of Γ that ﬁx a boundary point ξ ∈ ∂H . If Pξ acts semi-simply, then Pξ is either

ﬁnite or has a normal subgroup N of ﬁnite order with quotient Pξ/N a 1-crystallographic

group. If P∞ does not act semi-simply, it is either ﬁnite or a 2-crystallographic group.

82 Proof. By conjugation, we assume ξ is the point at inﬁnity. By passing to a subgroup of index 2 if necessary, we may assume that P∞ is orientation-preserving.

3 Note that we can denote a rotation about the vertical axis above the origin in H by a α 0 matrix for some α ∈ with |α| = 1. As a rotation about the axis above any 0 α−1 C 3 3 point z ∈ ∂H \ {∞} is just a rotation about the axis above 0 ∈ ∂H conjugated by the 1 z β z(β−1 − β) parabolic , a quick calculation gives this rotation as . 0 1 0 β−1

Suppose P∞ has no inﬁnite-order elements; then P∞ is a group of rotations about vertical

3 axes in the half-space model for H . Suppose P∞ contains two nontrivial rotations r and s about distinct vertical axes; without loss of generality we assume r rotates about the axis

above 0 and s rotates about the axis above some Z ∈ C\{0}. Then there exist α, β ∈ C\{1} with |α| = |β| = 1 and such that α 0 β z(β−1 − β) r = , s = . 0 α−1 0 β−1 Then the element 1 βz(β − β−1)(1 − α2) rsr−1s−1 = ∈ P 0 1 ∞ is parabolic, and thus contradicts the assumption that P∞ has no inﬁnite-order elements.

Thus, P∞ can only have nontrivial rotations about one axis, and the discreteness of Γ therefore guarantees that P∞ is ﬁnite.

Now suppose that P∞ does have inﬁnite-order elements; then by Proposition 3.34 they must either all be hyperbolic or parabolic.

If they are hyperbolic, we would like to show that all stabilize a particular geodesic.

0 Suppose not; let h and h be hyperbolic isometries in P∞ that are vertical screw motions about distinct vertical axes; by further conjugation, we may assume h ﬁxes the vertical axis

0 above 0 and h ﬁxes the axis above (x, y) = z ∈ C axis with z 6= 0. Then we may write

83 α 0 β z(β−1 − β) h = and h0 = for some α, β ∈ \{1} with |α| = |β| = 1. 0 α−1 0 β−1 C But this gives β cα−2n(β−1 − β) h−nh0hn = , 0 β−1

whose existence in P∞ ≤ Γ violates discreteness of Γ. Thus, all hyperbolic elements of P∞

stabilize a unique geodesic c(R). If r is a rotation in P∞, then r must rotate about (i.e., ﬁx) the vertical axis c that the hyperbolic elements stabilize, so that conjugation by r doesn’t

give hyperbolic elements that stabilize distinct axes. Thus P∞ ≤ Mc.

Consider the subgroup F ≤ Mc of isometries that ﬁx c(R) pointwise; F E Mc, and

F ≤ Γc(0) implies |F | < ∞. The quotient group Q = Mc/F has an eﬀective cocompact

action on c(R) ' E, and so satisﬁes the deﬁnition of a 1-crystallographic group. Similarly,

N = F ∩ Pξ is ﬁnite and normal in Pξ with the quotient acting eﬀectively and cocompactly

on c(R) ' E.

Now suppose all inﬁnite elements of P∞ are parabolic. Then all parabolic elements are of 1 α the form p = for some α ∈ . By discreteness of Γ, the parabolic elements (plus 0 1 C

I) correspond to a subgroup P ≤ P∞ isomorphic to a lattice in C via p 7→ α. Moreover, by 2 Claim 3.35 P is isomorphic to Z , so P is a cocompact lattice in C. ˆ 2 P∞ acts eﬀectively on C\{∞} ' E in the obvious way, so P∞ is itself 2-crystallographic.

As the above proof demonstrates that Pξ subgroups with hyperbolic elements are really

3 just Mc subgroups for some geodesic c(R) ⊂ H . We henceforth assume that Pξ is a subgroup that does not act semi-simply; in particular, Pξ does not as a group ﬁx any other boundary

3 point of H . We can now construct a collection adapted to the pair (FIN , VC).

3 Proposition 3.37. Let Γ ≤ Isom(H ) be a lattice, and let FIN and VC be the families of ﬁnite and virtually cyclic subgroups of Γ, respectively. The collection A of inﬁnite maximal

84 3 subgroups Mc that stabilize a geodesic c(R) ⊂ H and inﬁnite maximal parabolic subgroups 3 Pξ that ﬁx a unique boundary point ξ ∈ ∂H is adapted to the pair (FIN , VC).

Proof. We check the conditions directly.

1. Let Mc,Pξ ∈ A. Then their intersection is the set of elliptic elements that ﬁx ξ and

ﬁx c(R) pointwise; this group is contained in the stabilizer Γc(0), which is ﬁnite.

0 0 Next, suppose Mc,Mc0 ∈ A with c 6= c . Then Mc ∩ Mc0 stabilizes both c(R) and c (R). The elements can’t be hyperbolic by Corollary 1.24, so they must be elliptic. We can

characterize the possible elliptic elements by what they do to the endpoints of c(R) 0 and c (R). Let r ∈ Mc ∩ Mc0 ; if r fixes the endpoints of either geodesic, then it fixes that geodesic pointwise; again, there are only finitely many such elements in a discrete

Γ. If r swaps the endpoints of both geodesics, then it is a reﬂection through points

0 c(a) and c (b) for some a, b ∈ R, and therefore through a hemisphere centered on the 3 boundary ∂H or a vertical plane. Moreover, this hemisphere must be orthogonal to both geodesics simultaneously. For any given point on any given geodesic line, such a

hemisphere is uniquely determined. Thus at most one such reflection exists. Mc ∩ Mc0 is thus contained in a finite union of finite groups, and so is finite.

0 3 Finally, suppose Pξ,Pξ0 ∈ A with ξ, ξ ∈ ∂H distinct. Then again, things in the intersection ﬁx both boundary points, and are thus elliptic and ﬁx pointwise the unique

geodesic between ξ and ξ0; such a group is ﬁnite.

−1 −1 2. For any given γ ∈ Γ and any Mc,Pξ ∈ A, γMcγ = Mγ.c and γPξγ = Pγ.ξ are both in Γ and maximal, so A is conjugacy closed.

−1 3. Let Mc ∈ A and suppose γ ∈ NΓ(Mc); then for any hyperbolic h ∈ Mc γhγ ∈ Mc,

−1 so γhγ stabilizes both c(R) and γ.c(R). Then by Proposition 1.27 c(R) = γ.c(R), so

γ must stabilize c(R). We conclude that γ ∈ Mc, and thus NΓ(Mc) = Mc. A similar

argument implies that NΓ(Pξ) = Pξ. 85 4. Let H be an inﬁnite virtually cyclic subgroup of Γ. By the argument given in Claim 3.36,

since Γ is inﬁnite and discrete it must have inﬁnite elements. Let H be a virtually cyclic

subgroup of Γ and let h ∈ Γ be inﬁnite; then h is either hyperbolic or parabolic. If h

is hyperbolic, it stabilizes some geodesic c(R); by the argument in Proposition 1.27, so

does H. Then the subgroup Mc of elements of Γ that stabilize c(R) is clearly maximal with respect to that property, and is inﬁnite as it contains H. Now suppose h is

3 parabolic; then h ﬁxes a unique boundary point ξ ∈ ∂H . Again by the argument in

Proposition 1.27, H ﬁxes the same point. Let Pξ be the subgroup of Γ of elements that

ﬁx ξ; Pξ is clearly maximal with respect to this property, and is inﬁnite as it contains H.

It remains to obtain ﬁnite and virtually cyclic models for elements in A and a model for

EΓ.

3 Claim 3.38. Let Γ ≤ Isom(H ) be a lattice, Mc the maximal inﬁnite subgroup of isometries in Γ that stabilize a geodesic c(R). Then E is a model for EMc and a point is a model for

EMc.

Proof. As Mc is virtually cyclic by Proposition 1.28, EMc is clearly modeled by a point.

Let N be the subgroup of Mc that ﬁxes c(R) pointwise; then N is a normal subgroup

of Mc and is finite as it is a subgroup of Γc(0). Let Q = Mc/N, and consider the canonical map φ : Mc → Q. Q must be a 1-crystallographic group due to its action on c(R) ' E, so by Proposition 1.48, E is a model for EQ. By Proposition 1.3 E is then a model for EΓ, as finite subgroups of Γ are exactly the family generated by φ-lifts of finite subgroups of Q.

3 Claim 3.39. Let Γ ≤ Isom(H ) be a lattice, Pξ the maximal inﬁnite nonsemi-simple subgroup 3 2 of isometries in Γ that ﬁx a unique boundary point ξ ∈ ∂H . Then EPξ is modeled by E

and EPξ is given by Proposition 1.49. 86 Proof. By the argument in Claim 3.36, Pξ is 2-crystallographic. The result then follows from Propositions 1.48 and 1.49.

3 3 Claim 3.40. Let Γ ≤ Isom(H ) be a lattice; then H is a model for EΓ.

3 H Proof. Let H ≤ Γ be finite; then by Corollary 1.19 (H ) is contractible. Let H be infinite; then by the argument in Claim 3.36, Γ contains an element not of finite order. This element

3 3 H 3 cannot be elliptic, and so acts freely on H . Thus, (H ) is empty, and H is therefore a model for EΓ as claimed.

The above three claims and our collection A give us all we need to apply the construction of Lafont and Ortiz.

3 Proposition 3.41. Let Γ be a lattice in Isom(H ). Let A be the collection of inﬁnite maximal

Mc or Pξ subgroups of Γ. Let H be a complete set of representatives of the conjugacy classes within A, and consider the following cellular Γ-pushout:

a β 3 Γ ×H EH H H∈H

α a Γ ×H EH X. H∈H

Then X is a model for EΓ. In the above cellular Γ-pushout, we require either (1) α is the

disjoint union of cellular H-maps (H ∈ H), β is an inclusion of Γ-CW-complexes, or (2) α is the disjoint union of inclusions of H-CW-complexes (H ∈ H), β is a cellular Γ-map.

Proof. Apply Proposition 3.37 and Claims 3.38 to 3.40 to Proposition 2.1.

Proposition 3.42. Let M be a connected, oriented, ﬁnite-volume hyperbolic 3-manifold (that

3 is, with fundamental group a lattice Γ ≤ Isom(H )). Then gd(Γ) = 3.

87 Proof. The above model is 3-dimensional; thus, gd(Γ) ≤ 3. If Γ is nonuniform it contains

2 subgroups isomorphic to Z ; the result then follows by Theorem 1.50 and Proposition 1.3. Suppose Γ is a uniform lattice; we use Bredon cohomology to show that cd(Γ) = 3. We

3 showed in Claim 3.40 that H is a model for EΓ; thus, gd(Γ) ≤ 3. If Γ were not torsion-free, 3 3 its action on H would not be free. But H /Γ ' M is a manifold, so Γ cannot have ﬁxed 3 points in H ; we conclude that Γ is torsion-free. But this means that FIN —the family

of ﬁnite subgroups of Γ—is really just TR = {{1}}. By Remark 1.35, taking Z to be our coeﬃcients, we have that

3 ∼ 3 3 H (Γ; Z) = H (H /Γ),

the ordinary cohomology of M as a 3-dimensional CW -complex. By Poincar´eduality, 3 ∼ H (M) = H0(M). The 0-th ordinary homology group of a connected manifold is Z; thus, 3 H (Γ; Z) is nontrivial. In particular this gives that 3 ≤ cd(Γ) ≤ gd(Γ) ≤ 3, and thus cd(Γ) = gd(Γ) = 3.

Now consider the following exact sequence coming from the Mayer-Vietoris sequence for

Bredon cohomology. a 2 a 3 3 3 ... → H (H; Z) → H (H; Z) ⊕ H (Γ; Z) → H (Γ; Z) → ... H∈H H∈H 3 ∼ We have demonstrated that H (Γ; Z) = Z; since gd(H) = 1 and gd(H) = 2, we know that 2 ∼ 3 ∼ H (H; Z) = H (H; Z) = {0}; this reduces the above sequence to:

3 ... → 0 → Z → H (Γ; Z) → ....

3 3 By exactness, the map Z → H (Γ; Z) must be injective; therefore H (Γ; Z) is nontrivial. This gives that 3 ≤ cd(Γ) ≤ gd(Γ) ≤ 3, and thus gd(Γ) = 3 as claimed.

88 CHAPTER 4

COMPACT 3-MANIFOLD GROUP CLASSIFICATION

It is our ﬁnal goal to bound the geometric dimension of EΓ for any group Γ that is the funda-

mental group of a connected, closed, oriented 3-manifold. To that end, we remind the reader

of a few facts regarding manifolds, with an eye toward the prime and JSJ decompositions of

3-manifolds. A thorough treatment can be found in [Jac80a].

Deﬁnition. An n-manifold is a metric space such that any point has a neighborhood that

n is homeomorphic to a ball in R (the n is often dropped if it is understood). A Riemannian manifold X is a smooth manifold that admits a Riemannian metric. If the isometry group

Isom(X) acts transitively, we say X is homogeneous. If in addition X has a quotient of ﬁnite

volume, X is unimodular.A geometry is a simply-connected, homogeneous, unimodular

Riemannian manifold along with its isometry group. Two geometries (X, Isom(X)) and

(X0, Isom(X0)) are equivalent if Isom(X) ∼= Isom(X0) and there exists a diﬀeomorphism X → X0 that respects the Isom(X), Isom(X0) actions. A geometry (X, Isom(X)) (often abbreviated X) is maximal if there is no Riemannian metric on X with respect to which the isometry group strictly contains Isom(X). A manifold M is called geometric if there is a geometry X and discrete subgroup Γ ≤ Isom(X) with free Γ-action on X such that

M is diﬀeomorphic to the quotient X/Γ; we also say that M admits a geometric structure modeled on X. Similarly, a manifold with nonempty boundary is geometric if its interior is geometric.

It is a consequence of the uniformization theorem that compact surfaces (2-manifolds) 89 admit Riemannian metrics with constant curvature; that is, compact surfaces admit geomet-

2 2 2 ric structures modeled on S , E , or H . In three dimensions, we are not guaranteed constant curvature. Thurston demonstrated that in 3-dimensions there are eight maximal geometries

3 3 3 2 2 up to equivalence ([Sco83, Theorem 5.1]): S , E , H , S × E, H × E, PSLg 2(R), Nil, and Sol. Thanks to Thurston ([Thu97, Theorem 4.7.10]), we know which 3-dimensional geometries have lattices that are not uniform:

Theorem 4.1. Noncocompact coﬁnite groups of automorphisms of a Thurston geometry

3 2 exist only for H , H × E and PSLg 2(R) (out of the eight basic geometries). Any such group 2 of automorphisms of H × E acts also as a coﬁnite group of automorphisms of PSLg 2(R), and vice versa.

Deﬁnition. A closed n-manifold is an n-manifold that is compact with empty boundary. A connected sum of two n-manifolds M and N, denoted M#N, is a manifold created by removing the interiors of a smooth n-disc Dn from each manifold, then identifying the boundaries

n−1 n S . An n-manifold is nontrivial if it is not homeomorphic to S .A prime n-manifold is a nontrivial manifold that cannot be decomposed as a connected sum of two nontrivial

n n n-manifolds; that is, M = N#P for some n-manifolds N,P forces either N = S or P = S . 2 3 An n-manifold M is called irreducible if every 2-sphere S ⊂ M bounds a ball D ⊂ M.

Theorem 4.2. All orientable prime manifolds are irreducible with the exception of S1 × S2.

Given the above deﬁnitions, we can ‘factor’ a nontrivial closed oriented 3-manifold uniquely ([Mil62]):

Theorem 4.3 (Prime Decomposition). Let M be a closed, orientable nontrivial 3-manifold.

Then M = P1# ... #Pn where each Pi is prime. Furthermore, this decomposition is unique up to order and homeomorphism.

We would like to be able to claim that each prime closed oriented 3-manifold is modeled on one of the eight Thurston geometries; this is not quite true. Instead, some of the prime 90 manifolds will not be geometric, but will be decomposible along ‘incompressible’ tori into

pieces that are. We ﬁrst give the necessary deﬁnitions.

2 Deﬁnition. Let M be a 3-manifold and S be a compact surface, distinct from S , properly embedded in M. Let D2 be a disk properly embedded in M with ∂D2 ⊂ S. If ∂D2 does not

also bound a disk in S, we call D2 a compression disk for S and call S compressible; if no such compression disk exists, we call S incompressible.

Remark 4.4. There is an algebraic version of this concept as well; let S be a compact surface properly embedded in a 3-manifold M along a map ι : S → M; then ι induces a map

ι∗ : π1(S) → π1(M) by taking a loop in S and considering its embedding in M. If this map

is injective, we say that S is π1-injective (or algebraically incompressible).

Though these two concepts are intuitively similar, and every π1-injective surface is incompressible, in general the converse is not true. However, if S is a two-sided properly-embedded compact surface, then the two concepts are equivalent.

With a few more theorems, we can ﬁnally state Thurston’s Geometrization Conjecture; see the introduction of [BBM+10] for more details.

Theorem 4.5 (JSJ Decomposition). For a closed, irreducible, orientable 3-manifold M there exists a collection T ⊆ M of disjoint incompressible tori such that each component of M \ T is either atoroidal or a Seifert ﬁbered manifold. A minimal such collection T is unique up to isotopy.

Deﬁnition. A compact, orientable, irreducible 3-manifold is Haken if its boundary is nonempty or if it contains a closed essential surface.

Theorem 4.6 (Thurston’s Hyperbolization Theorem). Any atoroidal Haken 3-manifold is hyperbolic or Seifert ﬁbered.

Theorem 4.7 (Perelman). Let M be a closed, orientable, irreducible, atoroidal 3-manifold.

Then: 91 1. If π1(M) is ﬁnite, then M is spherical.

2. If π1(M) is inﬁnite, then M is hyperbolic or Seifert ﬁbered.

We now state the theorem:

Theorem 4.8 (Geometrization Conjecture). Every connected, closed, orientable, prime 3-

manifold can be cut along a (possibly empty) collection of tori so that the interiors of the

remaining submanifolds are geometric with ﬁnite volume.

We sketch how we can decompose any connected, closed, oriented 3-manifold M into geometric pieces. First, break M up along its prime decomposition into connected, closed, oriented, prime 3-manifolds; if any of these are atoroidal, then they are geometric by Theo-

1 2 rem 4.7. If they aren’t atoroidal, then in particular they aren’t S ×S , so they are irreducible. We can decompose them (Theorem 4.5) along a ﬁnite subset of the embedded tori into con-

nected, compact, oriented 3-manifolds, each of which is atoroidal or Seifert ﬁbered.

Unfortunately, the general JSJ decomposition occassionally goes too far; there are con-

nected, closed, oriented, prime manifolds called mapping tori that are given by T × [0, 1]/ ∼,

where the relation ∼ is deﬁned by (t, 0) ∼ (φ(t), 1) for some diﬀeomorphism φ : T → T. Upon

removing the essential embedded torus of such a mapping torus, one is left with T × (0, 1), which is not a ﬁnite-volume quotient as needed.

Fortunately, the mapping torus itself must be either Seifert ﬁbered or modeled on Sol,

so their existence does not contradict Thurston’s Geometrization Conjecture. Moreover,

this manifold T × (0, 1) can only appear in the JSJ decomposition of a connected, closed, oriented, prime 3-manifold M if M itself is a mapping torus; if the boundary tori were

embedded separately in M, they would be isotopic to one another along T × [0, 1] ⊆ M, violating our choice of minimal T .

Remark 4.9. When referring to the JSJ decomposition of a connected, closed, oriented,

prime manifold M, we will tacitly assume that M is not a mapping torus. As mapping tori 92 are either Seifert ﬁbered or modeled on Sol, no problems will arise in the general construction with this extra assumption.

We now take the models constructed in the previous chapter and discuss how to put them together along this decomposition. Of particular use to us is the following version of the Seifert-van Kampen theorem on a general graph of spaces (see [Kap01, Section 10.2]).

Deﬁnition. Suppose we are given a ﬁnite graph Y , and in the realization real(Y ) each edge y and vertex P is assigned a connected CW -complex Xy and XP , respectively. Additionally, suppose that we have closed cellular embeddings yP : Xy → XP if either P = o(y) or P = t(y). We assume that the images of these embeddings are disjoint. Glue the ends of the product space Xy × [0, 1] to the spaces XP using these embeddings; the resulting CW -complex is called a graph of spaces.

Let (Y, {XP ,Xy, yP }) be a graph of spaces; then this determines a graph of groups (G, Y ) where GP = π1(XP ) for each vertex P ∈ vert Y and Gy = π1(Xy) for each edge y ∈ edge Y . Moreover, we have the following generalization of Seifert-van Kampen:

Theorem 4.10 ([Kap01, Theorem 10.22]). The fundamental group of each graph of spaces is naturally isomorphic to the fundamental group of the corresponding graph of groups.

This result is useful in considering both decompositions of a connected, closed, oriented

2 3-manifold M; the prime decomposition corresponds to a graph of spaces with a copy of S associated to each edge and a prime submanifold to each vertex; an edge joins two vertices if

2 the two corresponding prime manifolds were connected along a copy of S in M. Similarly, given an irreducible such manifold M, the JSJ decomposition gives a graph space in which each edge is assigned a copy of T and each vertex is given a connected, compact, oriented submanifold, with an edge connecting two vertices if the submanifolds corresponding the the vertices share the torus corresponding to the edge in their boundaries.

93 We now give a slight twist on a construction of L¨uck and Weiermann ([LW12, Theorem

2.3]). Let M be a connected, closed, oriented 3-manifold decomposed along a disjoint collec-

2 tion S = {Sj}j∈J of incompressible surfaces (or copies of S ); let {Ni}i∈I be the connected, compact, oriented 3-manifolds whose disjoint union is the remainder M \S. Let G = π1(M), and let Gi = π1(Ni) and Hj = π1(Sj) for each i ∈ I or j ∈ J. We then create the graph of groups (G, Y ) associated to the graph of spaces M; to each vertex Pi ∈ vert Y associated to the manifold Ni we associate the group Gi, and to each edge yj ∈ edge Y associated to Sj we associate the group Hj.

Let Pi be a vertex in Y ; then we associate to Pi the family VCPi of virtually cyclic

subgroups of Gi. Given a model XPi for the classifying space EGi, take the Borel construction

G ×Gi XPi ; similarly, we construct Borel constructions G ×Hj Xyj associated to each edge yi ∈ edge Y .

Recall that the Borel construction G ×Hj Xyj is a disjoint union of copies of Xyj , one for

each left coset in G/Hj; by abuse of notation, let gXyj be the copy associated to the coset gHj. We wish to construct a CW -complex X from these pieces by gluing them together in the appropriate way—along edges of the Bass-Serre covering tree T . Let o(yj) = Pi in Y , and let y ∈ edge T be the edge associated to gH ; then if P ∈ vert T is the vertex e yj e ∼ |yj | Pe = o(y) = gog(y) = gGPi ∈ vert T , we have that Gyi = G ≤ GPi . So Xyj is a model for e |yj |

EGi by Corollary 1.4, and therefore the universal property of XPi gives a unique embedding a a Xyj → XPi ; we thus deﬁne the map α : G ×Hj Xyj → G ×Gi XPi by letting α inject edge Y vert Y gXyi into gXPj along this embedding. Similarly, if Qi is the terminal vertex t(yj), we deﬁne a a the map β : G ×Hj Xyj → G ×Gi XPi by injecting the copy gXyj associated to the edge Y vert Y coset gHyij into the copy ggyj XQi of XQi along the unique embedding Xyj → XQi given by

the universality of XQi . We can now make the following claim:

94 Proposition 4.11. Let M be a connected, closed, oriented 3-manifold with fundamental

group G = π1(M). Consider the following cellular G-pushout, as constructed above:

a α a G ×Hj Xyj G ×Gi XPi edge Y vert Y

β a G ×Gi XPi X. vert Y

The X is a model for EF G, where F is the family of virtually cyclic subgroups of G that are

conjugate to a virtually cyclic subgroup in one of the Gi or Hj.

Proof. Let φ : X → T be a deformation retraction collapsing each copy of XPi down onto

the vertex Pe ∈ vert T to which it corresponds, and similarly collapsing each Xyj ×[0, 1] down along the X component to the corresponding edge y ∈ edge T , y ' [0, 1]. G then has a yj e e natural action on X via left multiplication, which permutes the copies of each XPi or Xyj so that φ is a G-equivariant map. It remains to show that the G-CW -complex X is a model

for EF G. Suppose ﬁrst that H ≤ G is not in F; then either H does not conjugate into any vertex

subgroup or is not virtually cyclic. If H doesn’t conjugate into any vertex subgroup, then

H H T = ∅; in particular, H does not ﬁx any copy of XPi or Xyj × [0, 1] in X, so X = ∅. If H isn’t virtually cyclic, then even if the H-action on T does ﬁx some nonempty subgraph,

the H-action on any corresponding XPi or Xyj × [0.1] must have empty ﬁxed set; again, this implies that XH = ∅.

On the other hand, suppose H ∈ F; that is, H is virtually cyclic and conjugates into some

vertex subgroup Pi. Then H ﬁxes the copy of XPi in X corresponding to a ﬁxed vertex in

H T , and as H is virtually cyclic and XPi is a model for EGPi ,(XPi ) is not empty. Moreover, given any two vertices P and Q in vert T ﬁxed by the H-action, the unique geodesic path

c in T connecting P and Q must also be ﬁxed; in particular, T H is a connected subgraph

95 H of the tree T , so that T is itself a tree. Let ye ∈ edge T be fixed, and consider the copy X × [0, 1] with φ-image y; then the H-action on X has nonempty fixed set (X )H , so in yj e yj yj H H particular (Xyj ) ×[0, 1] is nonempty. Thus, the φ-preimage of T is a nonempty, connected subspace XH ⊆ X. To see that XH is contractible, first contract down along φ to T H , then contract the tree T H .

In particular, the above result allows us to construct models for EF G over either a prime decomposition or JSJ decomposition. From this model, we wish use the pushout construction

in Proposition 2.11 to build a model for EG.

We make the following two claims:

Proposition 4.12. Let M be a connected, closed, oriented 3-manifold, and suppose it has

prime decomposition P1# ... #Pn. Let G = π1(M), and let (G, Y ) be the graph of groups corresponding to the graph space given by the decomposition of M. Then the splitting of G as the fundamental group of (G, Y ) is acylindrical.

Proof. Let T be the Bass-Serre covering tree of (G, Y ), and let c be a path of length 1.

Then as G acts without inversion on T , elements that stabilize c must ﬁx the unique edge

ye ∈ edge T belonging to c. But the group associated to the edge y = p(ye) ∈ edge Y is 2 the fundamental group of S , and is therefore trivial. Any element that stabilizes ye must

be conjugate to an element in Hy = {1}, so the stabilizer group of c is trivial; the result follows.

Proposition 4.13. Let M be a connected, closed, irreducible, oriented 3-manifold, and let

(G, Y ) be the graph of groups associated to its minimal JSJ decomposition (minimal in that it uses the fewest embedded tori). Then the splitting of G as the fundamental group of (G, Y )

is acylindrical.

Proof. Let T be the Bass-Serre covering tree of (G, Y ), and let c be a path of length 3.

96 Again, G acts without inversion on T , so elements that stabilize c must in fact ﬁx it, or else they invert the center edge of c.

Let c have edges {ye1, ye2, ye3}, and let o(yei) = Pei−1, t(yei) = Pei in T . Let yi = p(yei) ∈ edge Y and Pi = p(Pei) ∈ vert Y for each vertex and edge in c. Let Ni be the manifolds that correspond to each vertex Pi, with fundamental groups Gi. Let Ti be the torus associated to each edge yi.

Now suppose that N1 is hyperbolic; then the subgroup Z1 ≤ G1 corresponding to the embedding of the fundamental group of T1 into G1 corresponds to a subgroup of parabolic

3 3 elements that ﬁx a boundary point ξ1 ∈ ∂H in the G1-action on H . Similarly, Z2 ≤ G1 corresponding to the embedding of T2 in G1 is a subgroup of parabolic elements that ﬁx some

3 ξ2 ∈ ∂H . Furthermore ξ1 6= ξ2, since the tori T1 and T2 correspond to different boundary components of N1. Any element g ∈ G that fixes ye1 and ye2 must lie in Z1 ∩ Z2 ≤ G1, and must therefore be a parabolic element that fixes both ξ1 and ξ2; by Corollary 1.26, there are no such nontrivial elements, so the group that fixes c must be trivial. A symmetric argument shows the same if N2 is hyperbolic.

Next, suppose both N1 and N2 are Seifert fibered with hyperbolic base orbifold, and consider the torus T2 that connects them. Note that N1 and N2 must have hyperbolic base manifolds B1 and B2, as noncompact Seifert fiber spaces with flat base manifolds that can appear in a JSJ decomposition have only one boundary torus (see the argument for

2 Proposition 3.10). The Gi act on H × R, with induced H2-action given by translation in 2 the R direction by one generator and a parabolic isometry on H in the other generator.

A nontrivial element g ∈ Γ that fixes c must fix two distinct boundary points in each Nfi; in particular, it fixes the boundary points associated with the embeddings of the tori into

Ni. As parabolic elements ﬁx precisely one boundary point by Corollary 1.26, this means g must act purely by an R-translation in Nfi; that is, g is supported by the ﬁbers of the

Seifert ﬁbration of each Ni. But this gives a Seifert ﬁber space structure to the manifold

97 N1 ∪ T2 ∪ N2, contradicting the minimality of the toral decomposition. Thus, it must be that there is no nontrivial element g ∈ Γ that ﬁxes c; in particular, the stabilizer of c is ﬁnite.

Suppose now that N1 is Seifert fibered with flat base orbifold B1; then N1 must be homeomorphic to the construction with fundamental group equal to that of the Klein bottle; none of the other Seifert fiber spaces with flat base orbifolds can appear in a minimal JSJ decomposition (cf. the argument preceding Proposition 3.10). N1 then has only one bounding torus, so it must be attached to precisely one other manifold in the graph of spaces, N2. As we have already dealt with the case when N2 is hyperbolic, it must be that N2 (and N0 = N2) is also Seifert fibered. If there exists a Siefert fiber structure on N2 whose induced structure on T2 lifts to a Seifert fiber structure of N1, then N1 ∪ T2 ∪ N2 has a Seifert fibered structure, contradicting the minimality of the JSJ decomposition; this happens if the G1-action on N1 induces a Z2-action on T2 that takes fibers to fibers. Suppose this doesn’t happen; then the induced fibrations of T1 ' T2 given by N1 and N2, respectivtely, are distinct and permuted by the Z2 action induced by N1. Let g ∈ Γ be a nontrivial element that fixes c. If N2 has hyperbolic base, then g must be supported by the fibers of N0 N2 as in the previous case.

2 2 Let E be the universal cover of the boundary torus of N1; then g acts on E via translation in the direction above both the ﬁbration induced by N0 and the ﬁbration induced by N2;

2 that is, g acts trivially on E . But the stabilizer of c embeds into the fundamental group of 2 this torus, so it acts eﬀectively on E ; this contradicts the supposition that g was nontrivial. We thus conclude that the stabilizer of c is ﬁnite.

The final case is when both N1 and N2 are Seifert fibered with flat base orbifolds B1 and

2 B2. Then the graph of groups (Γ0,Y ) is just a segment, with vertex groups both Z o Z2 2 and edge group Z that embeds in the vertex groups in the obvious way. As Γ is the direct ∼ 2 2 limit of this graph of groups, we have that Γ = Z o (Z2 ∗ Z2), where Z is the embedding of the edge group in Γ and the Z2 factors correspond to the embeddings of the Z2-factors in 2 the two vertex groups. As the Z2 factors have the same action on the normal Z subgroup,

98 ∼ 2 ∼ 3 we may write Γ = Z (D∞) = Z o Z2, where the additional Z factor corresponds to words ∼ 2 of even length in Z2 ∗ Z2 = D∞. The Z2 factor still acts effectively on the Z subgroup, 3 so it acts effectively on Z . Γ is therefore 3-crystallographic; in particular, Γ is Seifert fibered, contradicting the minimality of the JSJ decomposition. We conclude that a minimal

decomposition gives an acylindrical splitting of Γ with k = 3.

We can now state the main result.

Theorem 4.14. Let M be a connected, closed, oriented 3-manifold. Let M = P1# ... #Pk

1 2 be the prime decomposition of M into closed, irreducible manifolds and copies of S × S .

For each closed, irreducible manifold Pi, let Ti be a minimal collection of incompressible tori whose removal decomposes Pi into connected compact Seifert ﬁbered or hyperbolic manifolds.

Let Γ = π1(M) be the fundamental group of M. Then gd(Γ) ≤ 4; we can classify gd(Γ) by the first set of conditions satisfied in the list below (in particular, M satisfies at least one of

the following):

3 1. If any of the irreducible Pi are modeled on E , then gd(Γ) = 4;

2. If any of the irreducible Pi have a JSJ decomposition along a (nonempty, disjoint)

collection Ti of essential tori, then 3 ≤ gd(Γ) ≤ 4;

3 2 3. If any of the irreducible Pi are modeled on one of H , H × E, PSLg 2(R), Sol, or Nil, then gd(Γ) = 3;

4. If k > 2 or if M = P1#P2 with |π1(P1)| > 2, then gd(Γ) = 2;

5. If Γ is virtually cyclic, then gd(Γ) = 0.

Proof. First, we demonstrate a model for EΓ exists and is of dimension no more than 4.

Suppose ﬁrst that M is irreducible and atoroidal; then we know by Theorem 4.7 that M

is gometric, and have demonstrated models of dimension no more than 4 for fundamental

99 groups of geometric manifolds (Theorem 1.50, Corollary 3.23, and Propositions 3.28, 3.31

and 3.42).

Suppose now that M is irreducible but not geometric; then by Theorem 4.6, we can de-

compose M along disjoint essential tori into pieces with geometric or Seifert ﬁbered interior;

3 2 by Theorem 4.1 if they are geometric then they are modeled on H , PSLg 2(R), or H × E.

Let N1,...,Nk be the geometric pieces of M, with Gi = π1(Ni) the fundamental group

of Ni, 1 ≤ i ≤ k. Then we have models for each EGi of dimension 3 by Corollary 3.17 and Proposition 3.42. Let (Γ,Y ) be the graph of groups associated to the decomposition

of M; then if F is the family of virtually cyclic subgroups of Γ that conjugate into one of

the groups Gi, we have a model for EF Γ by Proposition 4.11; moreover, as the edge spaces

2 are of the form E × [0, 1] where E is a model for EZ , these models are 4-dimensional by Theorem 1.50. To obtain a model for EΓ, note that (Γ,Y ) gives an acylindrical splitting of

Γ by Proposition 4.13, and so we have a model for EΓ constructed in Proposition 2.11; this

model is 4-dimensional, as EF Γ is.

1 2 If M is prime but not irreducible, then by Theorem 4.2, M ' S × S . This is modeled 2 on S × E, and therefore has virtually cyclic fundamental group Γ. We know therefore that a point is a model for EΓ; thus gd(Γ) = 0 ≤ 4.

Finally, suppose M is not prime; then by Theorem 4.3, we can decompose M into prime

manifolds P1# ··· #Pm. Let Hi = π1(Pi) be the fundamental groups, 1 ≤ i ≤ m. By

the above work, we have a model for EHi of dimension no more than 4 for each prime manifold. Let (Γ,Y ) be the graph of groups associated to the prime decomposition; then

again by Proposition 4.11 we have a model for EF Γ of dimension no more than 4; note that the vertex spaces are now simply intervals [0, 1]. By Proposition 4.12 (Γ,Y ) gives an

acylindrical splitting of Γ, so by Proposition 2.11 we construct a model for EΓ; again, this

model is of dimension no more than 4.

Let us now attempt a ﬁner analysis of several cases to ﬁnish the proof.

100 3 1. If M has a prime submanifold P modeled on E , then Γ has a subgroup G = π1(P ) with gd(G) = 4 by Theorem 1.50. Then by Corollary 1.4, gd(G) ≤ gd(Γ) and therefore

gd(Γ) = 4.

2. If M has an irreducible prime manifold P in its prime decomposition with a nontrivial

decomposition along essential tori, then letting G = π1(P ), the model constructed for

3 EG is 4-dimensional. If no other prime manifold is modeled on E and no essential torus in P is the boundary of two hyperbolic manifolds, then there is no known subgroup G

of Γ for which gd(G) = 4; as such, we cannot claim precise knowledge of gd(Γ). We

2 do know that G contains a subgroup isomorphic to Z ; in particular, the fundamental 2 group of any essential embedded torus corresponds to such a subgroup. As gd(Z ) = 3 2 by Theorem 1.50, and as gd(Z ) ≤ gd(Γ) by Corollary 1.4, we know that 3 ≤ gd(Γ) ≤ 4 (see ‘Further Questions’ section for more details).

3. Suppose the prime decomposition of M consists only of connected, closed, irreducible,

1 2 atoroidal manifolds and copies of S ×S , and suppose further that none of the pieces are 3 modeled on E . Let (Γ,Y ) be the graph of groups corresponding to the prime decom-

position; then the construction in Proposition 4.11 for a model for EF Γ is dimension no more than 3, which means the model for EΓ constructed in Proposition 2.11 is also no

more than 3. In particular, we have gd(Γ) ≤ 3. On the other hand, if one of the prime

3 2 pieces is modeled on any of H , H × E, PSLg 2(R), Sol, or Nil, then the fundamental

group Gi of that piece is a subgroup of Γ with gd(Gi) = 3 by Propositions 3.28, 3.31

and 3.42 and Corollary 3.23. Thus, by Corollary 1.4, 3 = gd(Gi) ≤ gd(Γ) ≤ 3, so gd(Γ) = 3.

4. Suppose M satisﬁes none of the above conditions; then M has a prime decomposition

into prime manifolds, all of which are atoroidal and none of which is irreducible. This

101 means that M is the connected sum of a ﬁnite number of closed manifolds modeled on

3 2 either S or S × E.

If M has at least three prime components, then it must have as a sub-manifold either

2 P1#P2 (with at least one of the Pi modeled on S × E) or P1#P2#P3 with each sub- 3 manifold a (nontrivial) quotient of S . As such, Γ contains a subgroup isomorphic to

one of Z ∗ Z, Z ∗ Zp, or Zp ∗ Zq ∗ Zr (where p, q, r ≥ 2). We demonstrate that each of

these subgroups has a subgroup isomorphic to F2 (the free group on two generators) by ﬁnding two elements not of ﬁnite order, neither of which is a power of the other.

As subgroups of free groups are free, this will force these elements to generate F2.

The ﬁrst case is trivial. In the second case, let a generate the Z factor and let b be a −1 ∼ nontrivial element in the Zp factor; then ha, bab i = F2. In the ﬁnal case, let a, b, and −1 ∼ c be nontrivial elements in Zp, Zq, and Zr, respectively; then hab, cabc i = F2.

If M has a sub-manifold P1#P2 with |π1(P1)| > 2, then Γ has a subgroup isomorphic

to π1(P1) ∗ π1(P2); we again construct a subgroup isomorphic to F2. If π1(P1) has

an element a of order at least three, and if b is any nontrivial element of π1(P2), then 2 ∼ hab, a bi = F2. If π1(P1) has no element of order greater than two, it must have distinct 0 0 ∼ elements a, a of order 2; then hab, a bi = F2.

We have therefore shown that Γ contains a subgroup isomorphic to F2. gd(F2) = 2, so therefore 2 ≤ gd(Γ) by Corollary 1.4. On the other hand, the usual construction of

an EΓ via Proposition 4.11 and Proposition 2.11 gives a model of dimension no more

than 2, so it must be that gd(Γ) = 2.

5. If M satisﬁes none of the above conditions, then either M has fundamental group ∼ 2 3 Z2 ∗ Z2 = D∞ or M is modeled on E × S or S ; either has virtually cyclic fundamental group, so a point is a model for EΓ. In particular, gd(Γ) = 0.

102 4.1 Further Questions

It remains to classify the geometric dimension of 3-manifold groups that are the fundamental group Γ of a connected, closed, oriented, irreducible 3-manifold M with nontrivial

JSJ decomposition. We showed in Theorem 4.14 that 3 ≤ gd(Γ) ≤ 4 by constructing a

4-dimensional model for EΓ and a subgroup G ≤ Γ with gd(G) = 3, but have not yet characterized the manifolds with either dimension, if indeed both are possible. In this section, we demonstrate a computation that could in theory characterize the minimal dimension of these manifolds.

We would like to use Bredon cohomology to further analyze gd(Γ). Let (Γ,Y ) be the graph of groups associated to a decomposition of M, let Gi be the group associated to each

Pi ∈ vert Y (that is, Gi is the fundamental group of the manifold with boundary Ni in the associated graph of spaces), let F be the family of virtually cyclic subgroups that conjugate into one of the Gi, and let M ∈ Mod-OF Γ be a module. Then by [Chi76, Theorem 2] we have the following long exact sequence coming from the pushout construction in Proposition 4.11:

∗ ∗ ∗ ∗ ∂ Y 3 α Y 3 2 ι 4 ∂ Y 4 ... → H (Gi; M) → H (Z ; M) → HF (Γ; M) → H (Gi; M) → .... vert Y ye∈A vert Y 4 As gd(Gi) = 3 by Corollary 3.17 and Proposition 3.42, we know that H (Gi; M) = 0 for

∗ 4 ∗ ∗ any module M and any vertex Pi; this forces ∂ (HF (Γ; M)) = {0}, and thus im ι = ker ∂ =

4 ∗ Y 3 2 ∗ Y 3 2 ∗ HF (Γ; M). But im ι = H (Z ; M)/ ker ι = H (Z ; M)/ im α . Thus, we can make ye∈A ye∈A the following claim:

∗ Y 3 Y 3 2 Claim 4.15. The map α : H (Gi; M) → H (Z ; M) is surjective if and only if vert Y ye∈A 4 HF (Γ; M) is trivial.

Thus, if one were convinced gd(Γ) = 3, one way to demonstrate this would be to show

∗ that the map α is surjective for any module M ∈ Mod-OF Γ. This would mean that cdF (Γ) = 3; by Remark 2.12 we would then have cd(Γ) = gd(Γ) = 3. In particular, the following conjecture would be suﬃcient: 103 Conjecture 4.16. Let G be the fundamental group of a compact manifold M with nonempty

boundary in the (nontrivial minimal) decomposition along embedded tori of a connected,

closed, oriented, irreducible 3-manifold; that is, let G be the fundamental group of

• a compact Seifert ﬁbered manifold with a torus in the boundary (distinct from T×[0, 1]), or

• a compact manifold whose interior supports a noncompact, ﬁnite volume, hyperbolic

metric.

∼ 2 Let T be a torus in the boundary of M, and let H = Z be the embedding of the fundamental 3 3 2 group of the torus in G. Then the induced map H (G; M) → H (Z ; M) is surjective.

In the case of a Seifert ﬁbered compact submanifold M, we can go a bit further, as we

have G as an extension:

1 → Z → G → G0 → 1,

∼ 2 where G0 is the orbifold fundamental group of the base orbifold B of M. H = Z is similarly split with the same kernel, so we can use spectral sequences to compute the cohomology

groups. A spectral sequence is a sequence of arrays of modules whose limit is used to

compute graded algebraic invariants (e.g. homology and cohomology groups). Often the

idea is to use a breakdown of a group to compute algebraic invariants for the whole in terms

of the parts. A general discussion of spectral sequences can be found in [McC85]; we quickly

discuss the basics here.

Generally, we deﬁne the Ek-page of the spectral sequence to be an array of modules

p,q p,q p,q p,q Ek ; p, q ∈ Z (with Ek = 0 if p < 0 or q < 0) and a set of maps ∂k : Ek → p+k,q−(k−1) p+k,q−(k−1) p,q Ek such that ∂k ◦ ∂k = 0. We then deﬁne the Ek+1-page by setting p,q p,q p−k,q+(k−1) p,q Ek+1 = ker ∂k / im ∂k ; essentially, Ek+1 is the homology of the chain complex at p,q the Ek position. 104 p,q p,q p,q Note that Ek = Ek+1 for large enough k; in particular, if k > max{p, q + 1}, then ∂k p+k,q−(k−1) p−k,q+(k−1) p−k,q+(k−1) maps into Ek = 0 (since k − 1 > q) and ∂k has domain Ek = 0

(since k < p). Thus, it makes sense to deﬁne the limit E∞-page as the limit of each eventually

p,q p,q n ﬁxed sequence (Ek )k∈N in the (p, q) position. We then write E2 (G) ⇒ H (G) for some algebraic invariant Hn(G) if there is a ﬁltration

n 0 = Λ0 ≤ Λ1 ≤ ... ≤ Λn ≤ H (G),

∼ n−i,i where Λi+1/Λi = E∞ (G). In order to deﬁne a particular spectral sequence for a particular algebraic invariant and class of groups, we need to deﬁne the entries of a starting page (often

E2). In the case of Bredon cohomology, we have the following (more general) result ([MP02,

Theorem 5.1]):

Theorem 4.17. Let G be an extension of the form 1 → Z → G → G0 → 1. Let VC and

VC0 be the families of virtually cyclic subgroups of G and G0, respectively. Then

p,q p q p+q E2 = H (G0; HVC(−; M)) ⇒ H (G; M); that is, the E2-page deﬁned above converges to the Bredon cohomology of G in the virtually cyclics with coeﬃcients in the OG-module M.

q Note that the OG0-module HVC(−; M) takes the orbit space G0/H for some H ∈ VC0 to q the Z-module H (He; M), where He is the preimage of H in G. 3 2 With this deﬁnition in mind, let us look at the computation of H (Z ; M). We have that p,q 2 p q E2 (Z ) = H (Z; HVC(−; M)); but Z is virtually cyclic, so cd(Z) ≤ gd(Z) = 0 as the point p space {∗} is a model for EZ. Thus, H (Z; N) is trivial for any N ∈ OZ and p > 0, and p,q therefore E2 (Z) is trivial if p > 0. Moreover, this means that all boundary maps ∂ mapping 0,q p,q p,q out of Ek (Z) are zero maps for k ≥ 2, and therefore E∞ (Z) = E2 (Z) for all p, q ∈ Z (that is, the sequence collapses at the second term). In particular, this tells us that

3 2 0,3 2 0,3 2 0 3 H (Z ; M) = E∞ (Z ) = E2 (Z ) = H (Z; HVC(−; M)). 105 p,q Similarly, for G the fundamental group of the Seifert ﬁbered piece, we have that E2 (G) =

p q 2 H (G0; HVC(−; M)), where G0 is either virtually free or Z oZ2 (the latter if the base orbifold B is not hyperbolic). Unfortunately in this situation the sequence doesn’t generally collapse

at the second term.

The naturality of the spectral sequence gives us the following commutative diagram:

E0,3( 2) E0,3(G) 2 Z φ 2

=∼ i

0,3 2 0,3 E∞ (Z ) E∞ (G) = Λ1(G)

=∼

H3( 2; M) H3(G; M). Z α∗

∗ 0,3 0,3 2 Thus, α will be surjective if and only if the composition φ ◦ i : E∞ (G) → E2 (Z ) is surjective. The map i is an inclusion; in fact, i is easy to understand in two cases. If the orb ∼ base orbifold B of M is ﬂat, then we know G0 = π1 (B) = Z o Z2; this group is virtually 2 cyclic, so gd(G0) = 0. Now just like the case of Z , we have that

3 0,3 0,3 0 3 H (G; M) = E∞ (G) = E2 (G) = H (G0; HVC(−; M)),

p,q as all entries E2 (G) with p > 0 are trivial. Here i is an isomorphism, as is the map

3 0,3 ∗ H (G; M) → E∞ (G). Thus, understanding α is as simple as understanding the natural map φ.

If B is hyperbolic we aren’t so lucky; G0 is virtually a ﬁnitely-generated free group with

at least two generators, so gd(G0) = 2. The spectral sequence will collapse at the third term; in particular

0,3 0,3 ∼ 0,3 E∞ (G) = E3 (G) = ker ∂2 .

0,3 0,3 2,2 Since we can no longer assume that ∂2 : E2 (G) → E2 (G) has trivial range, we only know 0,3 ∼ 0,3 0,3 0,3 that E∞ (G) = ker ∂2 ≤ E2 (G); that is, that i is an inclusion with image ker ∂2 . Thus,

106 the following conjecture would be enough to prove (with one exeption) the ﬁrst case of the

previous conjecture:

Conjecture 4.18. Let M be a compact Seifert ﬁbered manifold with torus in the boundary and hyperbolic base orbifold B; let G0 be the orbifold fundamental group of B. Then for each

0 3 0 3 element x ∈ H (Z;H (−; M)), there exists an element x¯ ∈ H (G0;H (−; M)) with

• φ(¯x) = x, and

0,3 • ∂2 (¯x) = 0.

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