  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.
