{"id":{"repo_id":"cambridge","oai_identifier":"oai:www.repository.cam.ac.uk:1810/305052"},"canonical_url":"https://search.dev.ndltd.org/etd/cambridge/oai:www.repository.cam.ac.uk:1810/305052","repository":{"repo_id":"cambridge","name":"Cambridge University","base_url":"https://api.repository.cam.ac.uk/server/oai/request"},"display":{"title":"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.","abstract_html":"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 LK(G), we define a Leavitt graph-of-groups algebra LK(\\mathcal{G}) for $\\mathcal{G}$. We extend Leavitt path algebra results to LK(\\mathcal{G}), including uniqueness theorems describing homomorphisms out of LK(\\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 LK(\\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.","abstract_has_math":true,"creators":["Freeland, Richard"],"institution":"University of Cambridge","degree_name":"Doctor of Philosophy (PhD)","degree_level":"Doctoral","degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":["Brookes, Chris"],"committee_chairs":[],"committee_members":[],"year":2019,"date_issued":"2019-09-30","date_published":"2019-09-30","updated_at":"2026-07-22T22:24:17Z","subjects":["Algebra","group theory","representation theory","Thompson's groups","Hecke algebras"],"languages":["en"],"rights":[],"rights_urls":["https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/f8f98a33-4068-4efe-a41a-6c1c52979b13/download","https://www.rioxx.net/licenses/all-rights-reserved/"],"identifier_entries":[]},"links":{"outbound_url":"https://doi.org/10.17863/CAM.52134","outbound_label":"DOI","outbound_source":"dc:identifier.doi"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Brookes, Chris"]},{"key":"dc:contributor.sponsor","label":"Sponsor","values":["I was sponsored by the research council EPSRC for the first 3.5 years of my thesis."]},{"key":"dc:creator","label":"Author","values":["Freeland, Richard"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.issued","label":"Date","values":["2019-09-30"]},{"key":"dc:publisher.institution","label":"Dc Publisher Institution","values":["University of Cambridge"]},{"key":"dc:relation.isreferencedby.uri","label":"Dc Relation Isreferencedby URI","values":["https://www.repository.cam.ac.uk/handle/1810/305052"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"key":"dc:type.qualificationlevel","label":"Dc Type Qualificationlevel","values":["Doctoral"]},{"key":"dc:type.qualificationname","label":"Dc Type Qualificationname","values":["Doctor of Philosophy (PhD)"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Algebra","group theory","representation theory","Thompson's groups","Hecke algebras"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/f8f98a33-4068-4efe-a41a-6c1c52979b13/download","https://www.rioxx.net/licenses/all-rights-reserved/"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.doi","label":"DOI","values":["10.17863/CAM.52134"]},{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/bc313595-5df1-4bf4-9d1a-98c784aae769/download","https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/5cc4476a-5a7e-4919-bb35-b981398ca1fb/download"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["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. 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