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Relating Thompson's Group V to Graphs of Groups and Hecke Algebras

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Relating Thompson's Group V to Graphs of Groups and Hecke Algebras View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Apollo Relating Thompson's group V to graphs of groups and Hecke algebras Richard Leo Freeland Trinity College Department of Pure Mathematics and Mathematical Statistics University of Cambridge This dissertation is submitted for the degree of Doctor of Philosophy. September 2019 Declaration This thesis is the result of my own work and includes nothing which is the outcome of work done in collaboration except as declared in the Preface and specified in the text. It is not substantially the same as any that I have submitted, or, is being concurrently submitted for a degree or diploma or other qualification at the University of Cambridge or any other University or similar institution except as declared in the Preface and specified in the text. I further state that no substantial part of my dissertation has already been submitted, or, is being concurrently submitted for any such degree, diploma or other qualification at the University of Cambridge or any other University or similar institution except as declared in the Preface and specified in the text. It does not exceed the prescribed word limit for the relevant Degree Committee. 1 Abstract This thesis is in two main sections, both of which feature Thompson's group V , relating it to classical constructions involving automorphism groups on trees or to representations of symmetric groups. In the first section, we take G to be a graph of groups, which acts on its universal cover, the Bass-Serre tree, by tree automorphisms. Brownlowe, Mundey, Pask, Spielberg and Thomas con- structed a C∗-algebra for a graph of groups, writtten C∗(G), which bears many similarities to the C∗-algebra of a directed graph G. Inspired by the fact that directed graph C∗-algebras C∗(G) have algebraic analogues in Leavitt path al- gebras LK (G), we define a Leavitt graph-of-groups algebra LK (G) for G. We extend Leavitt path algebra results to LK (G), including uniqueness theorems describing homomorphisms out of LK (G), and establish a wider context for the algebras by showing they are Steinberg algebras of a particular ´etalegroupoid. Finally we show that certain unitaries in LK (G) form a group we can understand as a variant of Thompson's V , combining features of both Nekrashevych-R¨over groups and Matui's topological full groups of one-sided shifts. We prove finite- ness and simplicity results for these Thompson variants. The latter section of this thesis turns to representation theory. We briefly state some results about representations of V (due to Dudko and Grigorchuk) which we generalize to the new family of Thompson groups, including a discussion of representations of finite factor type and Koopman representations. Then, we describe how one would try to construct a Hecke algebra for V , built from copies of the Iwahori- Hecke algebra of Sn in a way inspired by how V can be constructed from copies of the symmetric group. We survey attempts to construct this and demon- strate what we believe to be the closest possible analogue to the Sn theory. We discuss how this construction could prove useful for understanding further representation theory. 2 Dedication In memory of Ben Elliott. 3 Acknowledgements Writing this thesis would not have been possible without the help of many people who deserve appreciation here. First, I would like to thank my supervisor Chris Brookes for introducing me to so many of the topics studied here; for his deep insight into which problems should be solvable; and for his continued patience and encouragement, without which I would have given up on research long ago. Thank you to my fellow algebra PhD students: Amit, Andreas, Nicolas and Stacey, for their friendship, the helpful mathematical conversations, and for their example to show me that finishing a PhD is possible. Thank you also to the EPSRC Research Council for my funding and to Trinity College for my accommodation during the last four yars. Finally I would like to thank my family - Kevin, Sue, Alex, and Katy - whose love and faith has been invaluable. 4 Chapter 1 Introduction The majority of this thesis is a study of groups acting on trees, or on their ends. One of the main purposes is to unite two different well-studied groups defined in terms of trees. The first is the fundamental group of Bass-Serre theory, which is an automorphism group of a tree defined in terms of a graph of groups G. The second is Thompson's group V and its many relatives, a group defined by permuting the ends of a binary tree, with many unusual properties. We will produce a family of groups combining features of both of these objects, and analyse it to see how properties of V and G transfer to these groups. In the final section of this thesis, we study representation theory of Thompson's group. We show how to generalize some known theorems about representations of V to the new family of Thompson-type groups. Finally, we discuss how the similarities between V and symmetric groups Sn might let us say more about representations of V , in particular trying as far as possible to extend a Hecke algebra construction from Sn to V . In this introduction we define the main objects of our consideration. We first describe the theory of automorphism groups of trees, which Bass and Serre described as fundamental groups of graphs of groups. Next, we exhibit a variety of related algebraic objects that act on paths in a directed graph, beginning with the Leavitt path algebra and finishing with Thompson's group V and its variants. We will combine both Leavitt path algebras and Thompson's groups with graphs of groups to form new objects. Notation: We begin by fixing some notation on directed graphs. A directed graph Γ is a quadruple (Γ0; Γ1; s; t) where Γ0 is a set of vertices,Γ1 is a set of 5 edges, and s and t are the source and target functions from Γ1 to Γ0. We say Γ is finite if both the vertex and edge sets are finite, and say Γ is locally finite if each of s−1(v) and t−1(v) are finite, for all v 2 Γ0. We say that a path in a directed graph Γ is either a vertex, or a sequence e1e2 : : : en of edges with s(ei) = t(ei+1) for 1 ≤ i ≤ n − 1 (so our paths are read right-to-left, like morphisms being composed). An example is shown in Figure 1.1. We write `(p) for the length of the path p, which is 0 if p is a vertex and n if p = e1e2 : : : en. Finally we extend the source and target maps to paths: we say that the path p = e1e2 : : : en has source s(p) = s(en) and target t(p) = t(e1), and if p is a vertex v, then s(p) = t(p) = v. Later we'll also consider infinite paths, which are infinite sequences of edges e1e2e3 ::: obeying the same condition s(ei) = t(ei+1). The target of an infinite path is defined as t(e1), but the source is not defined. e1 e2 en Figure 1.1: An example of a path We will write Γ∗ for the set of all (finite) paths, and Γn for the set of all ! paths e1e2 : : : en, of length n. We write Γ for the set of infinite paths. If K is a field, the path algebra KΓ is then the K-algebra with Γ∗ as basis, and with multiplication defined on the basis by concatenation of paths, wherever defined (i.e. for paths p and q, p · q is pq whenever pq is a path and 0 otherwise, when s(p) 6= t(q).) In this thesis, we will always specify when a graph is directed. The word `graph' alone will mean a graph in the sense of Serre (which is a directed graph with some additional structure), as given in the next section. 1.1 Bass-Serre theory We begin by studying automorphism groups of trees. The most important work in this area was developed by Serre and explained clearly in his book [51], and later developed by Bass in [8]. Our exposition is a combination of [51] and [13]. 1.1.1 Basic definitions Here we use Serre's notion of a graph. It is a particular kind of directed graph: 6 Definition 1.1.1. A graph Γ is a directed graph (Γ0; Γ1; s; t) equipped with a function − :Γ1 ! Γ1, written e 7! e¯, such that s(¯e) = t(e); t(¯e) = s(e), and e 6=e ¯ but e = e¯. This means that every edge e of a graph is directed, and comes with a reverse e¯; the edges can be partitioned into disjoint pairs fe; e¯g. We allow multiple edges between the same pair of vertices, and we allow loops whose source and target is the same edge. It may help to think of Serre's graphs as undirected graphs in the usual sense, but where every edge has been replaced by a pair of edges, oppositely directed. For v 2 Γ0, we define the degree (or valence) of v to be js−1(v)j, the cardi- nality of the set of edges with source v. We say the graph Γ is locally finite if all valencies are finite, and will usually work with locally finite graphs in this thesis. Since s(e) = t(¯e) for all edges e, we have jt−1(v)j = js−1(v)j, so locally finite graphs also have finitely many edges with target v for any vertex v.
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