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 $\mathcal{G}$ 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.

Name dc:type.qualificationname

Doctor of Philosophy (PhD)

Level dc:type.qualificationlevel

Doctoral

Grantor dc:publisher.institution

University of Cambridge

Year dc:date.issued

2019

Author dc:creator

• Freeland, Richard

Advisor dc:contributor.advisor

• Brookes, Chris

• Algebra
• group theory
• representation theory
• Thompson's groups
• Hecke algebras

Licence

• https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/f8f98a33-4068-4efe-a41a-6c1c52979b13/download
• https://www.rioxx.net/licenses/all-rights-reserved/

Language dc:language

en

DOI dc:identifier.doi

https://doi.org/10.17863/CAM.52134

OAI identifier oai:identifier

oai:www.repository.cam.ac.uk:1810/305052