Previously one of us introduced a family of groups $$G^M_L(S)$$
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 $$G^M_L(S)$$
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 $$G_L^M(S)$$
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.
We present a new method for showing that groups are virtually special. This is done by considering finite quotients and linear characters. We use this to show that an infinite family of groups, related to Bestvina-Brady groups and branching, provides new examples of virtually special groups outside of a hyperbolic context.
We construct an uncountable sequence of groups acting uniformly properly on hyperbolic spaces. We show that only countably many of these groups can be virtually torsion-free. This gives new examples of groups acting uniformly properly on hyperbolic spaces that are not virtually torsion-free and cannot be subgroups of hyperbolic groups.
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