
Random groups do not have Property (T) at densities below 1/4, Calum J. Ashcroft
We prove that random groups in the Gromov density model at density d≤ 1/4 do not have Property (T), answering a conjecture of Przytycki. We also prove similar results in the kangular model of random groups.
arXiv

Property (T) in random quotients of hyperbolic groups at densities above 1/3, Calum J. Ashcroft
We study random quotients of a fixed nonelementary hyperbolic group in the Gromov density model. Let G=⟨ST⟩ be a finite presentation of a nonelementary hyperbolic group, and let Ann_{ℓ,ω}(G)
be the set of elements of norm between ℓ−ω(ℓ) and ℓ in G. A random quotient at density d and length ωnear ℓ is defined by killing a uniformly randomly chosen set of S_{ℓ}(G)^{d}
words in Ann_{ℓ,ω}(G), where ω(ℓ)=o_{ℓ}(1). We prove that for any d>1/3, such a quotient has Property (T) with probability tending to 1 as ℓ tends to infinity. This result
answers a question of GromovOllivier and strengthens a theorem of Żuk (c.f KotowskiKotowski).
arXiv

Link conditions for the Haagerup property, Calum J. Ashcroft
We provide a condition on the links of a polygonal complex X that is sufficient to ensure Aut(X) has the Haagerup property, and hence so do any closed subgroups of Aut(X) (in particular, any group acting properly on X).
We provide an application of this work by considering the group of automorphisms of simplyconnected triangle complexes where the link of every vertex is isomorphic to the graph F090A, as constructed by Świątkowski.
arXiv

Property (T) in densitytype models of random groups, Calum J. Ashcroft
We study Property (T) in the Γ(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 kangular model of random groups,
i.e. the Γ(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 Γ(n,k,d) model has Property (T) with probability tending to 1 as k tends to infinity, strengthening
the results of Żuk and KotowskiKotowski, who consider only groups in the Γ(n,3k,d) model.
arXiv

On the eigenvalues of ErdösRényi random bipartite graph, Calum J. Ashcroft
We analyse the eigenvalues of ErdösRényi random bipartite graphs. In particular, we consider p satisfying n_{1}p=Ω(√n_{1}plog^{3}(n_{1})), n_{2}p=Ω(√n_{2}plog^{3}(n_{2})),
and let G∼G(n_{1},n_{2},p). We show that with probability tending to 1 as n_{1} tends to infinity:
μ_{2}(A(G))≤2[1+o(1)](√n_{1}p+√n_{2}p+√(n_{1}+n_{2})p).
arXiv

Link conditions for cubulation, Calum J. Ashcroft
We provide a condition on the links of polygonal complexes that is sufficient to ensure groups acting properly discontinuously and cocompactly on such complexes contain a virtually free codimension1 subgroup.
We provide stronger conditions on the links of polygonal complexes, which are sufficient to ensure groups acting properly discontinuously and cocompactly on such complexes act properly discontinuously on a
CAT(0) cube complex. 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 HaglundWise); 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].
arXiv

On random presentations with fixed relator length, Calum J. Ashcroft and Colva M. RoneyDougal
The standard (n, k, d) model of random groups is a model where the relators are chosen randomly from the set of cyclically reduced words of length k on an nelement generating set. Gromov’s density model of random groups
considers the case where n is fixed, and k tends to infinity. We instead fix k, and let n tend to infinity. We prove that for all k≥2 at density d ≥ 1/2 a random group in this model is trivial or cyclic of order
two, whilst for d≤ 1\slash 2 such a random group is infinite and hyperbolic. In addition, we show that for d≤ 1/ k such a random group is free, and that this threshold is sharp. These extend known results for
the triangular (k=3) and square (k=4) models of random groups.
arXiv ■ Comm. Algebra

On the average box dimensions of graphs of typical continuous functions, Bea AdamDay, Calum Ashcroft, Lars Olsen, Nicola Pinzani, Aluna Rizzoli, James Rowe
Let X be a bounded subset of ℝ^{𝑑} and write 𝐶𝗎(𝑋) for the set of uniformly continuous functions on X equipped with the uniform norm. Hyde et al. have recently proved that the box counting function of the graph
of a typical function 𝑓∈𝐶𝗎(𝑋) diverges in the worst possible way as 𝛿↘0. More precisely, Hyde et al. showed that for a typical function 𝑓∈𝐶𝗎(𝑋), the lower box dimension of the graph of f is as small as possible and if X
has only finitely many isolated points, then the upper box dimension of the graph of f is as big as possible. In this paper we will prove that the box counting function of the graph of a typical function 𝑓∈𝐶𝗎(𝑋) is
spectacularly more irregular than suggested by the result due to Hyde et al. Namely, we show the following surprising result: not only is the box counting function in divergent as 𝛿↘0, but it is so irregular that it remains
spectacularly divergent as 𝛿↘0 even after being “averaged" or “smoothened out" using exceptionally powerful averaging methods including all higher order Hölder and Cesàro averages and all higher order Riesz–Hardy logarithmic
averages. For example, if the box dimension of X exists, then we show that for a typical function 𝑓∈𝐶𝗎(𝑋), all the higher order lower Hölder and Cesàro averages of the box counting function are as small as possible, namely,
equal to the box dimension of X, and if, in addition, X has only finitely many isolated points, then all the higher order upper Hölder and Cesàro averages of the box counting function are as big as possible, namely, equal
to the box dimension of X plus 1.
Acta. Math. Hungar.

Cubulating CAT(0) groups and Property (T) in random groups PhD Thesis, University of Cambridge (2021), supervised by Henry Wilton.
Funded by EPSRC studentship 2114468.
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 m_{1}p = Omega(log m_{2}), and let G ~ G(m_{1}, m_{2}, p). We show that with probability tending to 1 as m_{1 }tends to infinity: μ_{2}(A(G))
≤O(sqrt{m_{2}p}). 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 kangular 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.
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