Title:

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

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 BassSerre tree, by tree automorphisms. Brownlowe, Mundey, Pask, Spielberg and Thomas constructed 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 algebras L_K(G), we define a Leavitt graphofgroups algebra L_K(G) for G. We extend Leavitt path algebra results to L_K(G), including uniqueness theorems describing homomorphisms out of L_K(G), and establish a wider context for the algebras by showing they are Steinberg algebras of a particular étale groupoid. Finally we show that certain unitaries in L_K(G) form a group we can understand as a variant of Thompson's V, combining features of both NekrashevychRöver groups and Matui's topological full groups of onesided 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 IwahoriHecke 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 demonstrate 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.
