Link conditions for cubulation

TL;DR

This paper establishes link conditions in polygonal complexes that guarantee groups acting on them have virtually special properties, leading to linearity and applications to specific classes of groups.

Contribution

It introduces new link conditions ensuring groups acting on polygonal complexes are virtually special, extending the class of groups known to have these properties.

Findings

Groups acting on complexes with certain link conditions are virtually special.

Groups from specific classes, like those acting on CAT(0) complexes with generalized quadrangle links, are virtually special.

The results imply linearity over Z for these groups.

Abstract

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].