We identify a class of "special" nonpositively curved cube complexes
that are closely related to right-angled Artin groups. We give
applications to subgroup separability and linearity, and to Coxeter
groups. Some sample consequence of our theory include:

1) Every word-hyperbolic Coxeter group has separable quasiconvex
subgroups.

2) Let G be the group given by the HNN extension . 
Then G is a subgroup of SL_n(Z) unless U and V have conjugate powers.

3) For each f.p. group Q, there is a short exact sequence 
1-> N -> G -> Q - > 1  where G < SL_n(Z) and N is f.g.

This is joint work with Frederic Haglund.