2026-09-21T08:01:57Zhttps://api.repository.cam.ac.uk/server/oai/request

oai:www.repository.cam.ac.uk:1810/3050522021-04-21T22:51:26Zcom_1810_213747com_1810_256064col_1810_213748

Relating Thompson's group V to graphs of groups and Hecke algebras 10.17863/CAM.52134 Freeland, Richard Brookes, Chris 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. University of Cambridge 2019-09-30 Thesis Doctoral Doctor of Philosophy (PhD) en I was sponsored by the research council EPSRC for the first 3.5 years of my thesis. https://www.repository.cam.ac.uk/handle/1810/305052 https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/bc313595-5df1-4bf4-9d1a-98c784aae769/download 6e799c7b66beb083bcd5c51ea4fbf99b https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/5cc4476a-5a7e-4919-bb35-b981398ca1fb/download b1acb97ee6c5d59df7b19250d2534983 https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/f8f98a33-4068-4efe-a41a-6c1c52979b13/download 87eda9de84448d1f82354d60eee3eb5f https://www.rioxx.net/licenses/all-rights-reserved/ Algebra group theory representation theory Thompson's groups Hecke algebras