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dc.contributor.authorFreeland, Richard
dc.date.accessioned2020-05-06T20:48:12Z
dc.date.available2020-05-06T20:48:12Z
dc.date.submitted2019-09-30
dc.identifier.urihttps://www.repository.cam.ac.uk/handle/1810/305052
dc.description.abstractThis 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.
dc.description.sponsorshipI was sponsored by the research council EPSRC for the first 3.5 years of my thesis.
dc.language.isoen
dc.rightsAll rights reserved
dc.subjectAlgebra
dc.subjectgroup theory
dc.subjectrepresentation theory
dc.subjectThompson's groups
dc.subjectHecke algebras
dc.titleRelating Thompson's group V to graphs of groups and Hecke algebras
dc.typeThesis
dc.type.qualificationlevelDoctoral
dc.type.qualificationnameDoctor of Philosophy (PhD)
dc.publisher.institutionUniversity of Cambridge
dc.publisher.departmentDPMMS
dc.date.updated2020-04-23T09:13:57Z
dc.identifier.doi10.17863/CAM.52134
dc.publisher.collegeTrinity
dc.type.qualificationtitleMaster of Mathematics
cam.supervisorBrookes, Chris
cam.thesis.fundingtrue


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