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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 By 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 2 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 He2H a Γ × EH X He e He2H 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 H2H a G ×H {∗} X H2H 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 decom- position of M conjectured by Thurston. In particular, the Seifert-van Kampen Theorem tells us that a decomposition of a man- ifold 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 decom- position 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.
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