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

Abstract

Previously one of us introduced a family of groups G^M_L(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^M_L(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.