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Free Groups, Raags, and Their Automorphisms

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Free Groups, Raags, and Their Automorphisms Free groups, RAAGs, and their automorphisms Schriftliche Habilitationsleistung vorgelegt von Dawid Kielak aus Gdu´nsk Juni 2017 Universit¨atBielefeld Universität Bielefeld Contents Introduction1 Free groups1 Automorphisms of free groups3 Automorphisms of RAAGs9 Free groups 13 Chapter I. Groups with infinitely many ends are not fraction groups 15 1. Introduction 15 2. The result 15 Chapter II. A characterisation of amenability via Ore domains 19 1. Introduction 19 2. The result 19 Automorphisms of free groups 21 Chapter III. Alexander and Thurston norms, and the Bieri{Neumann{Strebel invariants for free-by-cyclic groups 23 1. Introduction 23 2. Preliminaries 25 3. The invariants for descending HNN extensions of free groups 37 4. Thurston, Alexander and higher Alexander norms 42 5. The L2-torsion polytope and the BNS-invariant 48 6. UPG automorphisms 53 Chapter IV. Nielsen Realisation by Gluing: Limit Groups and Free Products 57 1. Introduction 57 2. Relative Stallings' theorem 59 3. Blow-ups 64 4. Relative Karrass{Pietrowski{Solitar theorem 68 5. Fixed points in the graph of relative free splittings 69 6. Fixed points in the outer space of a free product 70 7. Relative Nielsen realisation 71 8. Nielsen realisation for limit groups 73 Chapter V. On the smallest non-abelian quotient of Aut(Fn) 77 1. Introduction 77 2. Preliminaries 79 3. Alternating groups 83 4. Sporadic groups 92 5. Algebraic groups and groups of Lie type 92 6. Groups of Lie type in characteristic 2 95 7. Classical groups in odd characteristic 101 8. The exceptional groups of Lie type 108 iii iv CONTENTS 9. Small values of n and the conclusion 111 10. Computations 113 Chapter VI. The 6-strand braid group is CAT(0) 119 1. Introduction 119 2. Definitions and preliminaries 121 3. Turning points and turning faces 127 4. Proof of the main theorem 130 5. The orthoscheme complex of a modular complemented lattice is CAT(0)139 Automorphisms of RAAGs 143 Chapter VII. Outer actions of Out(Fn) on small RAAGs 145 1. Introduction 145 2. The tools 147 3. The main result 155 4. From larger to smaller RAAGs 159 Chapter VIII. Nielsen realisation for untwisted automorphisms of right-angled Artin groups 161 1. Introduction 161 2. Preliminaries 163 3. Relative Nielsen Realisation 169 4. Systems of subgraphs and invariance 171 5. Cubical systems 175 6. Cubical systems for free products 179 7. Gluing 180 8. Proof of the main theorem: preliminaries 186 9. Proof of the main theorem: part I 187 10. Proof of the main theorem: part II 195 Bibliography 209 Introduction The research presented in this thesis revolves around the structure of free groups, their close relatives the right-angled Artin groups, and the automorphisms of groups in both classes. As such, it positions itself inside the areas of combina- torial and geometric group theory. The former originated in the early 20th century in the work of Dehn, and was further developed by Higman, Magnus, Nielsen, and Whitehead, to name just a few. The theory evolved with time and in its later stages was heavily reshaped by the work of Stallings. In the 1980s Mikhail Gromov revo- lutionised the subject, and geometric group theory was born. This latter branch of mathematics is concerned with studying groups via their actions on metric spaces; this point of view proved to be extremely fruitful. As in the case of every theory, there are objects that geometric group theory is particularly focused on. Initially, these were the classes of hyperbolic and CAT(0) groups, as well as mapping class groups of surfaces, which connect geometric group theory to Thurston-style geometric topology. More recently, we observe a shift of interest from mapping class groups to the groups of automorphisms of free groups (most notably Out(Fn), the group of outer such automorphisms); it is worth point- ing out that automorphism groups of free groups as well as mapping class groups were an active area of research in the earlier days of combinatorial group theory. The solution of the Virtually Fibred Conjecture of Thurston due to Agol [Ago] (based on the work of Wise) brought right-angled Artin groups (RAAGs) into focus. These groups are given by presentations in which the only relators are commutators of some generators. Hence RAAGs interpolate between free and free-abelian groups, and so their outer automorphism groups form a natural class of groups interpolating between Out(Fn) and GLn(Z), the latter being an important group of more classical interest. Free groups Free groups form a cornerstone of group theory, and are objects of central importance in combinatorial and geometric group theory. The structure of a free group is extremely simple (indeed, as simple as possible for a group), and hence most questions one can ask about a group have immediate answers for free groups. This is however not the case for all questions { there are non-trivial results about the structure of free groups. One such result (of significance for the current discussion) is the fact (essentially due to Magnus) that free groups are orderable, that is one can endow a free group F with a total order invariant under left multiplication. In fact, free groups are biorderable, that is the total order can be taken to be invariant under left and right multiplication simultaneously. The orderability of Fn, the free group on n generators, features crucially in the proof of the Hanna Neumann Conjecture due to Mineyev [Min]. When a group G is orderable, the set of all possible orderings can be topologised in the following way: to each ordering 6 we associate the positive cone P6 = fx 2 G j x > 1g 1 2 INTRODUCTION This allows us to view the set of orderings as a subset of the power set of Fn given the product topology, thus becoming the space of orderings (note that the topology of this space is referred to as the Sikora topology). For Fn this space has been shown to be homeomorphic to a Cantor set { this follows from the work of McCleary [McC], but appears in this form for the first time in an article of Navas [Nav]. Finding the homeomorphism type of the corresponding space of biorderings of Fn is still an open problem. Observe that the positive cone P6 is a subsemigroup of G satisfying −1 G = P6 t P6 t f1g −1 −1 where P6 = fx j x 2 P6g, and t denotes the disjoint union. In particular, −1 G = P6P6 , that is every element in G can be written as a fraction of two elements in P6 (we say that G is the group of fractions of its subsemigroup P6). One important feature of the Sikora topology is that any positive cone in G finitely generated as a semigroup is isolated in the space of orderings. Thus, observ- ing that the space of left orderings of Fn has no isolated points (as it is homeomor- phic to the Cantor set), we deduce that P6 is not finitely generated as a semigroup for any left-invariant ordering 6 on Fn. One can investigate the general structure of subsemigroups P ⊆ Fn of which Fn is a fraction group. Navas conjectured that such a P cannot at the same time be finitely generated (as a semigroup) and satisfy P \ P −1 = ;. ChapterI contains the proof of this conjecture; in fact much more is proved. TheoremI.2.3. Let P be a finitely generated subsemigroup of a finitely generated group F with infinitely many ends. If PP −1 = F then P = F . Note that Fn is a particular example of a group with infinitely many ends. The proof uses the topology of a group with infinitely many ends directly, but let us mention here a celebrated result of Stallings [Sta1, Sta2]: a finitely generated group has at least 2 ends if and only if it splits over a finite subgroup, that is acts on a tree without a global fixed point and with a single edge-orbit with a finite stabiliser. In the language of Bass{Serre theory, such a group is the fundamental group of a non-trivial graph of groups with exactly one edge and a finite edge group. Moreover, the group in question has infinitely many ends if and only if it is additionally not virtually cyclic. We will return to Stallings' theorem when discussing automorphisms of free groups. Coming back to TheoremI.2.3, as an application we obtain a new proof of the fact that the space of orderings of Fn is homeomorphic to the Cantor set. It is worth noting that the proof offered in ChapterI is the first geometric one. We also deduce that the orderings of finitely generated groups with infinitely many ends do not have finitely generated positive cones. This was already known for free products of orderable groups by the work of Rivas [Riv]. Free groups, interesting as they intrinsically are, perform also an extrinsic func- tion in group theory. It is often useful to establish whether a given group G contains a free subgroup; this is especially important when studying amenability, as being amenable and possessing F2 as a subgroup are mutually exclusive properties. The von Neumann{Day Conjecture claimed that in fact possessing F2 was the only obstruction to amenability. The conjecture has been proven wrong by Olshanskiy in 1980, but still groups which are not amenable and do not contain F2 are considered to be somewhat exotic. Amenability is an extremely interesting property of a group. One of the def- initions of amenability of a (discrete) group G is the statement that G admits a left-multiplication invariant mean, but amenability has many other equivalent definitions which are more natural in specific contexts.
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