On the virtual and residual properties of a generalization of Bestvina-Brady groups

Abstract

Previously one of us introduced a family of groups $G_L^M(S)$, parametrized by a finite flag complex $L$, a regular covering $M$ of $L$, and a set $S$ of integers. We give conjectural descriptions of when $G_L^M(S)$ is either residually finite or virtually torsion-free. In the case that $M$ is a finite cover and $S$ is periodic, there is an extension with kernel $G_L^M(S)$ and infinite cyclic quotient that is a CAT(0) cubical group. We conjecture that this group is virtually special. We relate these three conjectures to each other and prove many cases of them.

arXiv comment23 pages, minor changes in version 2