Relating Thompson's group V to graphs of groups and Hecke algebras

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 $\mathcal{G}$ 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 constructed a $C^$-algebra for a graph of groups, writtten $C^(\mathcal{G})$, 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 algebras $L_K(G)$, we define a Leavitt graph-of-groups algebra $L_K(\mathcal{G})$ for $\mathcal{G}$. We extend Leavitt path algebra results to $L_K(\mathcal{G})$, including uniqueness theorems describing homomorphisms out of $L_K(\mathcal{G})$, and establish a wider context for the algebras by showing they are Steinberg algebras of a particular '{e}tale groupoid. Finally we show that certain unitaries in $L_K(\mathcal{G})$ form a group we can understand as a variant of Thompson's $V$, combining features of both Nekrashevych-R"{o}ver groups and Matui's topological full groups of one-sided shifts. We prove finiteness 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 $\mathfrak{S}_n$ in a way inspired by how $V$ can be constructed from copies of the symmetric group. We survey attempts to construct this and demonstrate what we believe to be the closest possible analogue to the $\mathfrak{S}_n$ theory. We discuss how this construction could prove useful for understanding further representation theory.