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Cohomological Finiteness Properties of Groups University of Southampton Faculty of Social and Human Sciences School of Mathematics Cohomological Finiteness Properties of Groups arXiv:1410.4363v1 [math.GR] 16 Oct 2014 Simon St. John-Green Supervisor: Prof. Brita Nucinkis Thesis for the degree of Doctor of Philosophy October 2014 University of Southampton Abstract Faculty of Social and Human Sciences School of Mathematics Doctor of Philosophy Cohomological Finiteness Properties of Groups by Simon St. John-Green The main objects of study in this thesis are cohomological finiteness conditions of discrete groups. While most of the conditions we investigate are algebraic, they are inspired by topological invariants, particularly those concerning proper actions on CW-complexes. The first two chapters contain preliminary material necessary for the remain- der of the thesis. Chapter 2 concerns modules over a category with an emphasis on finiteness conditions. This material is well-known for (EI) categories, but we use a more general setup applicable to Mackey and cohomological Mackey func- tors, needed in Chapter 4. Chapter 3 specialises to Bredon cohomology, giving an overview of some results and detailing a few interesting examples. In Chapter 4 we study finiteness conditions associated to Bredon cohomology with coefficients restricted to Mackey functors and cohomological Mackey func- tors, building again on the material in Chapter 2. In particular we characterise the corresponding FPn conditions and prove that the Bredon cohomological di- mension with coefficients restricted to cohomological Mackey functors is equal to the F-cohomological dimension for all groups. We prove in Chapter 5 that for groups of finite F-cohomological dimension, the F-cohomological dimension equals the Gorenstein cohomological dimension, and give an application to the behaviour of the F-cohomological dimension under group extensions. If a group G admits a closed manifold model for BG then G is a Poincar´e duality group, in Chapter 6 we study Bredon{Poincar´eduality groups, a gener- alisation of these. In particular if G admits a cocompact manifold model X for H EFin G (the classifying space for proper actions) with X a submanifold for any finite subgroup H of G, then G is a Bredon{Poincar´eduality group. We give several sources of examples, including using the reflection group trick of Davis to produce examples where the dimensions of the submanifolds XH are specified. We classify Bredon{Poincar´eduality groups in low dimensions and examine their behaviour under group extensions. In Chapter 7 we study Houghton's group Hn, calculating the centralisers of virtually cyclic subgroups and the Bredon cohomological dimension with respect to both the family of finite subgroups and the family of virtually cyclic subgroups. Contents List of Figures ix Declaration of Authorship xi Acknowledgements xiii Chapter 1. Introduction 1 1.1. Free actions and group cohomology 1 1.2. Proper actions and Bredon cohomology 2 1.3. Modules over a category 4 1.4. nG and F-cohomological dimension 5 1.5. Mackey and cohomological Mackey functors 7 1.6. Gorenstein cohomological dimension 9 1.7. Bredon duality groups 11 1.8. Houghton's groups 14 Chapter 2. Modules over a category 17 2.1. Tensor products 21 2.1.1. Tensor product over C 21 2.1.2. Tensor product over R 22 2.2. Frees, projectives, injectives and flats 22 2.3. Restriction, induction and coinduction 24 2.4. Tor and Ext 26 2.5. Finiteness conditions 28 Chapter 3. Bredon modules 31 3.1. Free modules 31 3.2. Restriction, induction and coinduction 34 3.3. Bredon homology and cohomology of spaces 37 3.4. Homology and cohomology of groups 38 3.5. Cohomological dimension 38 3.5.1. Low dimensions 39 3.6. FPn conditions 40 3.6.1. Quasi-OF FPn conditions 41 3.7. Change of rings 42 v vi CONTENTS 3.8. Some interesting examples 44 3.9. Finitely generated projectives and duality 47 Chapter 4. Mackey and cohomological Mackey functors 55 4.1. Introduction 56 4.1.1. Mackey functors 56 4.1.1.1. Free modules 60 4.1.1.2. Induction 61 4.1.1.3. Homology and cohomology 62 4.1.2. Cohomological Mackey functors 63 4.1.2.1. Explicit description of π 66 4.1.2.2. Homology and cohomology 67 4.2. FPn conditions for Mackey functors 68 4.3. Homology and cohomology of cohomological Mackey functors 72 4.4. FPn conditions for cohomological Mackey functors 77 4.4.1. HF FPn implies FFPn 79 4.4.2. FFPn implies HF FPn 81 4.5. Cohomological dimension for cohomological Mackey functors 83 4.5.1. Closure properties 85 4.6. The family of p-subgroups 86 4.6.1. FPn conditions over Fp 90 Chapter 5. Gorenstein cohomology and F-cohomology 93 5.1. Preliminaries 93 5.1.1. Complete resolutions and complete cohomology 93 5.1.2. F-cohomology 94 5.1.3. Complete F-cohomology 95 5.1.4. Gorenstein cohomology 96 5.2. FG-cohomology 97 5.2.1. Construction 97 5.2.2. Technical results 99 5.2.3. An Avramov{Martsinkovsky long exact sequence in F-cohomology101 5.3. Group extensions 105 5.4. Rational cohomological dimension 105 Chapter 6. Bredon duality groups 107 6.1. Preliminary observations 108 6.2. Examples 109 6.2.1. Smooth actions on manifolds 110 6.2.2. A counterexample to the generalised PDn conjecture 113 6.2.3. Actions on R-homology manifolds 114 CONTENTS vii 6.2.4. One relator groups 116 6.2.5. Discrete subgroups of Lie groups 118 6.2.6. Virtually soluble groups 118 6.2.7. Elementary amenable groups 119 6.3. Finite extensions of right-angled Coxeter groups 120 6.4. Low dimensions 124 6.5. Extensions 127 6.5.1. Direct products 127 6.5.2. Finite-by-duality groups 129 6.6. Graphs of groups 131 6.7. The wrong notion of Bredon duality 134 Chapter 7. Houghton's groups 137 7.1. Centralisers of finite subgroups in Hn 138 7.2. Centralisers of elements in Hn 142 7.3. Brown's model for EFin Hn 147 7.4. Finiteness conditions satisfied by Hn 149 7.4.1. The pushout of L¨uck and Weiermann 150 7.4.2. Calculation of OVCyccd Hn 152 Bibliography 155 Index 163 List of Figures 1 A graphical representation of the set Sa described in Section 7.1. 142 ix Declaration of Authorship I, Simon St. John-Green, declare that this thesis and the work presented in it are my own and has been generated by me as the result of my own original research. Title: Cohomological Finiteness Properties of Groups I confirm that: (1) This work was done wholly or mainly while in candidature for a research degree at this University; (2) Where any part of this thesis has previously been submitted for a degree or any other qualification at this University or any other institution, this has been clearly stated; (3) Where I have consulted the published work of others, this is always clearly attributed; (4) Where I have quoted from the work of others, the source is always given. With the exception of such quotations, this thesis is entirely my own work; (5) I have acknowledged all main sources of help; (6) Where the thesis is based on work done by myself jointly with others, I have made clear exactly what was done by others and what I have contributed myself; (7) Either none of this work has been published before submission, or parts of this work have been published as: (a) Centralisers in Houghton's groups (2012, to appear Proc. Edinburgh Math. Soc.) [SJG12]. (b) Bredon{Poincar´eduality groups (2013, to appear J. Group Theory) [SJG13a]. (c) Finiteness conditions for Mackey and cohomological Mackey func- tors (J. Algebra 411 (2014), no. 0, 225{258) [SJG14] (d) On the Gorenstein and F-cohomological dimensions (2013, to ap- pear Bull. Lond. Math. Soc.) [SJG13b]. Signed: Date: October 2014. xi Acknowledgements First and foremost I'd like to thank my supervisor Brita Nucinkis, for intro- ducing me to this beautiful area of mathematics and for being an endless source of motivation, support, and guidance throughout my studies. I'd like to thank my advisor Ian Leary for his support and for many useful mathematical insights, including the examples in Section 6.3 and Example 6.2.11. I thank him as well for reading a first draft of this thesis. For making the last three years such a wonderful experience, I thank all the PhD students and post-docs, past and present, from the Southampton mathemat- ics department, including but not limited to: Alex Bailey, Matt Burfitt, Valerio Capraro, Chris Cave, Yuyen Chien, Charles Cox, Dennis Dreesen, Michal Ferov, Robin Frankhuizen, Tom Harris, Pippa Hiscock, Ana Khukhro, Dave Matthews, Ingrid Membrillo, Raffaele Rainone, Amin Saied, Conor Smyth, Rob Snocken, Joe Tait, Edwin Tye, and Mike West. The material in Sections 7.4.1 and 7.4.2 is joint work with Nansen Petrosyan and I thank him for suggesting the topic and for sharing with me some fascinating mathematics. Thanks to Dieter Degrijse, not only for pointing out to me an error in Lemma 3.9.5 while reading a preprint of [SJG14], but also for many interesting conver- sations concerning Mackey and cohomological Mackey functors. I'd like to thank Conchita Mart´ınez-P´erezfor her very helpful comments concerning a preprint of [SJG12], material now contained in Chapter 7. Thanks to the referee of the Proceedings of the Edinburgh Mathematical Society who reviewed the same preprint for making useful comments, including suggesting the graph Γ used in Section 7.2.
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