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We associate cube complexes called completions to each subgroup of a right-angled Coxeter group (RACG). 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 RACG. We provide an algorithm that determines whether a given one-ended, 2-dimensional RACG is isomorphic to some finite-index subgroup of another given RACG. In addition, we answer several algorithmic questions regarding quasiconvex subgroups. Finally, we give a new proof of Haglund’s result that quasiconvex subgroups of RACGs are separable.
Cite this article
Pallavi Dani, Ivan Levcovitz, Subgroups of right-angled Coxeter groups via Stallings-like techniques. J. Comb. Algebra 5 (2021), no. 3, pp. 237–295DOI 10.4171/JCA/54