On strongly quasiconvex subgroups

TL;DR

This paper develops a comprehensive theory of strongly quasiconvex subgroups in finitely generated groups, generalizing hyperbolic group concepts and exploring their properties across various group classes.

Contribution

It introduces the concept of strong quasiconvexity, characterizes it via divergence, and analyzes its behavior in hyperbolic, relatively hyperbolic, Coxeter, and Artin groups.

Findings

Strong quasiconvexity is preserved under quasi-isometry.

Characterization of strong quasiconvexity via lower relative divergence.

Complete descriptions in relatively hyperbolic and right-angled Coxeter groups.

Abstract

We develop a theory of \emph{strongly quasiconvex subgroups} of an arbitrary finitely generated group. Strong quasiconvexity generalizes quasiconvexity in hyperbolic groups and is preserved under quasi-isometry. We show that strongly quasiconvex subgroups are also more reflexive of the ambient groups geometry than the stable subgroups defined by Durham-Taylor, while still having many analogous properties to those of quasiconvex subgroups of hyperbolic groups. We characterize strongly quasiconvex subgroups in terms of the lower relative divergence of ambient groups with respect to them. We also study strong quasiconvexity and stability in relatively hyperbolic groups, right-angled Coxeter groups, and right-angled Artin groups. We give complete descriptions of strong quasiconvexity and stability in relatively hyperbolic groups and we characterize strongly quasiconvex special subgroups and stable special subgroups of two dimensional right-angled Coxeter groups. In the case of right-angled Artin groups, we prove that two notions of strong quasiconvexity and stability are equivalent when the right-angled Artin group is one-ended and the subgroups have infinite index. We also characterize non-trivial strongly quasiconvex subgroups of infinite index (i.e. non-trivial stable subgroups) in right-angled Artin groups by quadratic lower relative divergence, expanding the work of Koberda-Mangahas-Taylor on characterizing purely loxodromic subgroups of right-angled Artin groups.