Link Conditions for Cubulation

Abstract

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 $\mathbb {Z}$, 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.