## Cubulating CAT(0) groups and Property (T) in random groups

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## 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 and n tends to infinity. We also prove that for d > 1/3, a random group in the (n, k, d) model has Property (T) with probability tending to 1 as k tends to infinity, strengthening the results of Zuk and Kotowski–Kotowski, who consider only groups in the (n, 3k, d) model.