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Profinite Properties of 3-Manifold Groups

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Profinite Properties of 3-Manifold Groups Profinite Properties of 3-Manifold Groups Gareth Wilkes St John's College University of Oxford A thesis submitted for the degree of Doctor of Philosophy Trinity 2018 Dedicated to my family. Acknowledgements Firstly I give great thanks to Marc Lackenby for his excellent supervision and advice throughout my DPhil and for being a font of knowledge from whom I have learnt a great deal. I would also like to thank him for so carefully reading all my mathematics and rooting out any misconceptions. I would also like to acknowledge several academics and professors who have invited me for visits over the course of my DPhil and with whom I have had very enlightening discussions. In particular I thank Henry Wilton at Cambridge, Stefan Freidl at Regensburg and Michel Boileau at Marseille. I also thank Peter Neumann for his advice on the giving of seminars and on proper mathematical style, as well as for illuminating Kinderseminar with his great knowledge. Huge thanks go to my Mum, my Dad, my sister Lizzie and all my family and friends for all their support over the last few years and for picking me up when I needed it. Among my friends I would particularly like to mention Joseph Mason, In´esDawson, Alexander Roberts, Federico Vigolo, Robert Kropholler, TomaˇsZeman, Josephine French, David Hume, Ric Wade and Guy-Andr´eYerro. Federico gets additional thanks for helping me with some of the diagrams. Finally I thank the plant Camellia sinensis for providing the lifeblood of my mathematics. Abstract In this thesis we study the finite quotients of 3-manifold groups, con- cerning both residual properties of the groups and the properties of the 3-manifolds that can be detected using finite quotients of the fundamental group. A key theme is the analysis of when two 3-manifold groups can have the same families of finite quotients. We make a detailed study of this ‘profinite rigidity' problem for Seifert fibre spaces and prove complete classification results for these manifolds. From Seifert fibre spaces we continue on this trajectory and extend our classification results to all graph manifolds. We illustrate this classification with examples and several consequences, including for graph knots and for mapping class groups. The third part of the thesis concerns the behaviour of the finite p-group quotients of 3-manifold groups. In general these quotients may be scarce and poorly behaved. We give results showing that some of these issues may be resolved by passing to finite-sheeted covers of the manifold involved. We also prove theorems concerning the p-conjugacy separability of certain graph manifold groups. The concluding chapter of the thesis collects other results linking low- dimensional topology and finite quotients of groups. In particular we prove that finite quotients of a right-angled Artin group distinguish it from other right-angled Artin groups, and we give an argument detecting the prime decomposition of certain 3-manifold groups from the finite p- group quotients. Statement of Originality Chapters 2 and 3 constitute an exposition of background material, written by the author but drawing on the sources cited within. Chapters 1, 4, 5 and 6 and Section 7.2 are entirely the author's own work. Section 7.1 was written by the author in collaboration with Robert Kropholler. To the best of my knowledge all citations are complete and accurate. Contents 1 Introduction 1 2 Preliminaries 8 2.1 Inverse limits and profinite completions . .8 2.1.1 Inverse limits . .8 2.1.2 Profinite completions . 10 2.1.3 The profinite topology . 14 2.1.4 The pro-p topology . 16 2.2 Cohomology of profinite groups . 17 2.2.1 Goodness and cohomological dimension . 18 2.2.2 Spectral sequence . 22 2.2.3 Chain complexes . 23 3 Profinite Bass-Serre Theory 28 3.1 Groups acting on profinite graphs . 28 3.1.1 Profinite graphs . 28 3.1.2 Profinite trees . 30 3.1.3 Group actions on profinite trees . 33 3.2 Acylindrical actions . 36 3.3 Graphs of profinite groups . 39 3.3.1 Free profinite products . 39 3.3.2 Graphs of groups . 42 i 4 Seifert Fibre Spaces 46 4.1 Background on Seifert fibre spaces . 46 4.2 Profinite completions of 2-orbifold groups . 50 4.3 Profinite rigidity of Seifert fibre spaces . 52 4.3.1 Preservation of the fibre . 54 4.3.2 Central extensions . 58 4.3.3 Action on cohomology . 61 4.4 Distinguishing Seifert fibre spaces among 3-manifolds . 71 5 JSJ Decompositions and Graph Manifolds 76 5.1 Background on JSJ decompositions . 76 5.2 The JSJ decomposition . 78 5.3 Graph manifolds versus mixed manifolds . 91 5.4 Totally hyperbolic manifolds . 92 5.5 The pro-p JSJ decomposition . 94 5.6 Profinite rigidity of graph manifolds . 98 5.7 Examples . 120 5.8 Commensurability of graph manifolds . 125 5.9 Behaviour of knot complements . 127 5.10 Relation to mapping class groups . 134 6 Virtual Pro-p Properties 141 6.1 Virtual p-efficiency . 141 6.2 Conjugacy p-separability . 149 7 Miscellany 164 7.1 RAAGs and special groups . 164 7.2 Pro-p prime decompositions . 170 ii Chapter 1 Introduction It has long been known that one can often use the finite quotients of a given group to obtain information about that group. Mal'cev observed that a finitely-presented group G such that every element is non-vanishing in some finite quotient (i.e. such that G is residually finite) has solvable word problem. The conjugacy and membership problems have similar answers given conditions ensuring a plentiful supply of finite quotients. Such conditions may be referred to by the catch-all name of `residual finiteness properties' or sometimes simply `residual properties'. For convenience one often assembles all the finite quotients of a group G into a single algebraic object Gb called the profinite completion of G and attempts to study ‘profinite invariants'|that is, those properties of G which can be recovered from the profinite completion. Of course in the broadest scope there is no hope of obtaining any significant invariants for the class of all groups, since taking a direct product or extension by an infinite simple group changes G without changing the finite quotients. One therefore looks to find classes of groups within which these problems are tractable. Those groups arising in low-dimensional topology often have good residual proper- ties. Work of Hempel [Hem87] together with Geometrization shows that all 3-manifold groups are residually finite, and the dramatic advances resulting from the theory of special cube complexes yield many powerful results. In dimension two, even the iso- morphism type of a Fuchsian group (among lattices in connected Lie groups) is a profinite invariant [BCR16]. Profinite invariants and residual properties of 3-manifold groups have received much recent attention. Lackenby [Lac14] showed that the property of containing a 1 pair of disjoint non-separating surfaces is detected by the pro-2 completion of the fundamental group of a 3-manifold. Wilton and Zalesskii [WZ17a] showed that the geometry of a 3-manifold is a profinite invariant. Bridson and Reid [BR15] and Boileau and Friedl [BF15b] showed that the fundamental group of the figure-eight knot complement is uniquely determined among 3-manifold groups by its finite quotients. This was later extended to all once-punctured torus bundles by Bridson, Reid and Wilton [BRW17]. These examples are ‘profinitely rigid' in the following sense. Definition 2.1.9. An (orientable) 3-manifold is profinitely rigid if the profinite com- pletion distinguishes its fundamental group from all other fundamental groups of (orientable) 3-manifolds. In the other direction, Funar [Fun13] building on Stebe [Ste72] found Sol manifolds which are not profinitely rigid; Hempel [Hem14] has found Seifert-fibred examples. We now give an overview of the thesis, and state some of our key results. In Chapter 2 we recall the basic notions of the study of profinite groups, and in particular the cohomology theory of profinite groups. None of this chapter is new, except for the results of Section 2.2.3 which are adapted from similar results in [Nak94]. Chapter 3 gives an overview of the theory of profinite groups acting on profinite trees developed mainly by Ribes and Zalesskii. The primary source for this chapter is the book [Rib17]. The results of Section 3.2 are probably known to experts but are not well- attested in the literature. In Chapter 4 we study profinite completions of Seifert fibre space groups. We prove the following strong theorem, in which the examples of Seifert fibre spaces which are not profinitely rigid are those found by Hempel [Hem14]. We also give a new proof that these examples do indeed fail to be rigid, and prove a version of Theorem A for Seifert fibre spaces with boundary. These results first appeared in the author's paper [Wil17b]. Theorem A. Let M be a closed orientable Seifert fibre space. Then either: • M is profinitely rigid; or 2 • M has the geometry H2 × R, and is a surface bundle with periodic monodromy φ; and in the latter case the 3-manifolds whose fundamental groups have the same finite k quotients as π1M are precisely the surface bundles with monodromy φ , for k coprime to the order of φ. In Chapter 5, which is adapted from the author's papers [Wil18a] and [Wil18b], we investigate 3-manifolds with a non-trivial JSJ decomposition. We pay particular at- tention to the case of graph manifolds. We obtain the following classification theorem for graph manifold groups with isomorphic profinite completions: Theorem B.
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