Malnormality and join-free subgroups in right-angled Coxeter groups

TL;DR

This paper investigates the properties of malnormal subgroups in right-angled Coxeter groups, establishing their quasiconvexity, join-freeness, and hyperbolic embedding, with implications for subgroup divergence and graph characterizations.

Contribution

It proves that finitely generated malnormal subgroups are strongly quasiconvex and join-free, and explores their divergence and hyperbolic embeddings in right-angled Coxeter groups.

Findings

Malnormal subgroups are strongly quasiconvex in one-ended right-angled Coxeter groups.

Infinite proper malnormal subgroups are join-free and virtually free.

Characterization of almost malnormal parabolic subgroups via defining graphs.

Abstract

In this paper, we prove that all finitely generated malnormal subgroups of one-ended right-angled Coxeter groups are strongly quasiconvex and they are in particular quasiconvex when the ambient groups are hyperbolic. The key idea is to prove all infinite proper malnormal subgroups of one-ended right-angled Coxeter groups are join-free and then prove the strong quasiconvexity and the virtual freeness of these subgroups. We also study the subgroup divergence of join-free subgroups in right-angled Coxeter groups and compare them with the analogous subgroups in right-angled Artin groups. We characterize almost malnormal parabolic subgroups in terms of their defining graphs and also recognize them as strongly quasiconvex subgroups by the recent work of Genevois and Russell-Spriano-Tran. Finally, we discuss some results on hyperbolically embedded subgroups in right-angled Coxeter groups.