Characterizations of stability via Morse limit sets

Garcia, Jacob

doi:10.2140/agt.2025.25.5541

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Characterizations of stability via Morse limit sets

Jacob Garcia

Algebraic & Geometric Topology 25 (2025) 5541–5566

DOI: 10.2140/agt.2025.25.5541

Abstract

Subgroup stability is a strong notion of quasiconvexity that generalizes convex cocompactness in a variety of settings. In this paper, we characterize stability of a subgroup by properties of its limit set on the Morse boundary. Given $H < G$ , both finitely generated, $H$ is stable exactly when all the limit points of $H$ are conical, or equivalently when all the limit points of $H$ are horospherical, as long as the limit set of $H$ is a compact subset of the Morse boundary for $G$ . We also demonstrate an application of these results in the settings of the mapping class group for a finite-type surface, $Mod (S)$ .

Keywords

Morse boundary, stable subgroup, quasiconvex, convex cocompact, mapping class group

Mathematical Subject Classification

Primary: 20F65, 20F67, 20F69

References

Publication

Received: 15 November 2023

Revised: 3 November 2024

Accepted: 30 December 2024

Published: 18 December 2025

Authors

Jacob Garcia
Mathematical Sciences Smith College Northampton, MA United States

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Volume 25, issue 9 (2025)