Boundary convex cocompactness and stability of subgroups of finitely generated groups

TL;DR

This paper explores the relationship between boundary convex cocompactness and subgroup stability in finitely generated groups, providing a new boundary-based characterization that generalizes classical concepts from hyperbolic and Kleinian groups.

Contribution

It introduces an equivalent boundary-based characterization of subgroup stability, extending classical notions to a broader class of groups using Morse boundary techniques.

Findings

Established boundary characterization of subgroup stability

Unified concepts across hyperbolic, Kleinian, and mapping class groups

Generalized convex cocompactness via Morse boundary

Abstract

A Kleinian group $\Gamma < \mathrm{Isom}(\mathbb H^3)$ is called convex cocompact if any orbit of $\Gamma$ in $\mathbb H^3$ is quasiconvex or, equivalently, $\Gamma$ acts cocompactly on the convex hull of its limit set in $\partial \mathbb H^3$. Subgroup stability is a strong quasiconvexity condition in finitely generated groups which is intrinsic to the geometry of the ambient group and generalizes the classical quasiconvexity condition above. Importantly, it coincides with quasiconvexity in hyperbolic groups and convex cocompactness in mapping class groups. Using the Morse boundary, we develop an equivalent characterization of subgroup stability which generalizes the above boundary characterization from Kleinian groups.