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

TL;DR

This paper introduces a new method using cube complexes called completions to analyze subgroups of right-angled Coxeter groups, providing tools for classification, algorithmic decision-making, and proofs of subgroup properties.

Contribution

It develops a novel completion technique for RACG subgroups, enabling characterization, algorithmic detection of subgroup isomorphisms, and a new proof of subgroup separability.

Findings

Reflection subgroups are quasiconvex.

One-ended Coxeter subgroups of 2D RACGs are quasiconvex.

An algorithm determines subgroup isomorphism in RACGs.

Abstract

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.