Cubulating CAT(0) groups and Property (T) in random groups
type: CreativeWork
description: This thesis considers two properties important to many areas of mathematics: those of cubulation and Property (T). Cubulation played a central role in Agol’s proof of the virtual Haken conjecture, while Property (T) has had an impact on areas such as group theory, ergodic theory, and expander graphs. The aim is to cubulate some examples of groups known in the literature, and prove that many ‘generic’ groups have Property (T). Graphs will be central objects of study throughout this text, and so in Chapter 2 we provide some definitions and note some results. In Chapter 3, we provide a condition on the links of polygonal complexes that allows us to cubulate groups acting properly discontinuously and cocompactly on such complexes. If the group is hyperbolic then this action is also cocompact, hence by Agol’s Theorem the group is virtually special (in the sense of Haglund–Wise); in particular it is linear over Z. We consider some applications of this work. Firstly, we consider the groups classified by [KV10] and [CKV12], which act simply transitively on CAT(0) triangular complexes with the minimal generalized quadrangle as their links, proving that these groups are virtually special. We further apply this theorem by considering generalized triangle groups, in particular a subset of those considered by [CCKW20]. To analyse Property (T) in generic groups, we first need to understand the eigenvalues of some random graphs: this is the content of Chapter 4, in which we analyse the eigenvalues of Erdös–Rényi random bipartite graphs. In particular, we consider p satisfying m1p = (log m2), and let G ~ G(m1, m2, p). We show that with probability tending to 1 as m1 tends to infinity: μ2(A(G)) <=O(sqrt{m2p}). In Chapter 5 we study Property (T) in the (n, k, d) model of random groups: as k tends to infinity this gives the Gromov density model, introduced in [Gro93]. We provide bounds for Property (T) in the k-angular model of random groups, i.e. the (n, k, d) model where k is fixed …
abstract: This thesis considers two properties important to many areas of mathematics: those of cubulation and Property (T). Cubulation played a central role in Agol’s proof of the virtual Haken conjecture, while Property (T) has had an impact on areas such as group theory, ergodic theory, and expander graphs. The aim is to cubulate some examples of groups known in the literature, and prove that many ‘generic’ groups have Property (T). Graphs will be central objects of study throughout this text, and so in Chapter 2 we provide some definitions and note some results. In Chapter 3, we provide a condition on the links of polygonal complexes that allows us to cubulate groups acting properly discontinuously and cocompactly on such complexes. If the group is hyperbolic then this action is also cocompact, hence by Agol’s Theorem the group is virtually special (in the sense of Haglund–Wise); in particular it is linear over Z. We consider some applications of this work. Firstly, we consider the groups classified by [KV10] and [CKV12], which act simply transitively on CAT(0) triangular complexes with the minimal generalized quadrangle as their links, proving that these groups are virtually special. We further apply this theorem by considering generalized triangle groups, in particular a subset of those considered by [CCKW20]. To analyse Property (T) in generic groups, we first need to understand the eigenvalues of some random graphs: this is the content of Chapter 4, in which we analyse the eigenvalues of Erdös–Rényi random bipartite graphs. In particular, we consider p satisfying m1p = (log m2), and let G ~ G(m1, m2, p). We show that with probability tending to 1 as m1 tends to infinity: μ2(A(G)) <=O(sqrt{m2p}). In Chapter 5 we study Property (T) in the (n, k, d) model of random groups: as k tends to infinity this gives the Gromov density model, introduced in [Gro93]. We provide bounds for Property (T) in the k-angular model of random groups, i.e. the (n, k, d) model where k is fixed …
author: Ashcroft, Calum James
language: eng
keywords: Group theory, Cube complexes, Random groups, Property (T)
license: https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/8ba036bb-d704-4f6f-9f02-0a2e716aedc7/download
license: https://creativecommons.org/licenses/by/4.0/
date published: 2021-09-16
date modified: 2026-07-22T22:24:23Z
provider: Cambridge University
publisher: Networked Digital Library of Theses and Dissertations
type: WebPage
source organization: University of Cambridge

PropertyValue
property ID: DOI
value: 10.17863/CAM.75926