We provide a condition on the links of polygonal complexes that is sufficiesnt 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].

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 $\mathbb {Z}$, virtually torsion free, and has the Haagerup property. We consider some applications of this work. Firstly, we consider the groups classified by Kangaslampi–Vdovina and Carbone–Kangaslampli–Vdovina, which act simply transitively on the vertices of $CAT(0)$ triangular complexes with the generalized quadrangle of order $2$ 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 Caprace–Conder–Kaluba–Witzel.

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 n-element 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 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.

No abstract