Search

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 k-angular model of random groups., Comment: 18 pages

We study random quotients of a fixed non-elementary hyperbolic group in the Gromov density model. Let $G=\langle S\;\vert\; T\rangle $ be a finite presentation of a non-elementary hyperbolic group, and let $Ann_{l,\omega }(G)$ be the set of elements of norm between $l-\omega(l)$ and $l$ in $G$. A random quotient at density $d$ and length $\omega$-near $l$ is defined by killing a uniformly randomly chosen set of $\vert S_{l}(G)\vert ^{d}$ words in $Ann_{l,\omega (l)}(G)$, where $\omega (l) =o_{l}(l)$. 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 Gromov--Ollivier and strengthens a theorem of \.{Z}uk (c.f Kotowski--Kotowski)., Comment: 39 pages. V2: corrected statement and proof of Theorem A, introduced Theorem B, added appropriate reference for Lemma 4.4

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 simply-connected triangle complexes where the link of every vertex is isomorphic to the graph F090A, as constructed by \'Swi\k{a}tkowski., Comment: 21 pages, V2: typos correctex. arXiv admin note: text overlap with arXiv:2012.09019

We study Property (T) in the $\Gamma(n,k,d)$ 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 $ \Gamma (n,k,d)$ model where $k$ is fixed and $n$ tends to infinity. We also prove that for $d>1\slash 3$, a random group in the $\Gamma(n,k,d)$ model has Property (T) with probability tending to $1$ as $k$ tends to infinity, strengthening the results of \.{Z}uk and Kotowski--Kotowski, who consider only groups in the $\Gamma (n,3k,d)$ model., Comment: 34 pages. V2 incorporates referee's suggestions. This is the final version accepted for publication

We analyse the eigenvalues of Erd\"os--R\'enyi random bipartite graphs. In particular, we consider $p$ satisfying $n_{1}p=\Omega(\sqrt{n_{1}p}\log^{3}(n_{1})),$ $n_{2}p=\Omega(\sqrt{n_{2}p}\log^{3}(n_{2})),$ and let $G\sim G(n_{1},n_{2},p)$. We show that with probability tending to $1$ as $n_{1}$ tends to infinity: $$\mu_{2} (A(G))\leq 2[1+o(1)](\sqrt{n_{1}p}+\sqrt{n_{2}p}+\sqrt{(n_{1}+n_{2})p}).$$, Comment: 5 pages; comments welcome

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 codimension-1 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 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]., Comment: 43 pages, 4 figures; v2: typos corrected