Right-angled mock reflection and mock Artin groups

We define a right-angled mock reflection group to be a group $G$ acting combinatorially on a CAT(0) cubical complex such that the action is simply-transitive on the vertex set and all edge-stabilizers are $\mathbb{Z}_2$. We give a combinatorial characterization of these groups in terms of graphs with local involutions. Any such graph $\Gamma$ not only determines a mock reflection group, but it also determines a right-angled mock Artin group. Both classes of groups generalize the corresponding classes of right-angled Coxeter and Artin groups. We conclude by showing that the standard construction of a finite $K(\pi,1)$ space for right-angled Artin groups generalizes to these mock Artin groups.