Subgroups of right-angled Coxeter groups via Stallings-like techniques

We associate a cube complex to any given finitely generated subgroup of a right-angled Coxeter group, called the completion of the subgroup. A completion characterizes many properties of the subgroup such as whether it is quasiconvex, normal, finite-index or torsion-free. We use completions to show that reflection subgroups are quasiconvex, as are one-ended Coxeter subgroups of a 2-dimensional right-angled Coxeter group. We provide an algorithm that determines whether a given one-ended, 2-dimensional right-angled Coxeter group is isomorphic to some finite-index subgroup of another given right-angled Coxeter group. In addition, we answer several algorithmic questions regarding quasiconvex subgroups. Finally, we give a new proof of Haglund's result that quasiconvex subgroups of right-angled Coxeter groups are separable.