Characterizations of Stability via Morse Limit Sets

TL;DR

This paper characterizes subgroup stability in terms of properties of limit sets on the Morse boundary, providing criteria for stability based on conical and horospherical limit points, with applications to mapping class groups.

Contribution

It offers a new characterization of subgroup stability via Morse boundary limit sets, linking geometric properties to algebraic stability in various groups.

Findings

Stability of a subgroup is equivalent to all limit points being conical.

All limit points being horospherical also characterizes stability.

Results apply to mapping class groups of finite type surfaces.

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, $\text{Mod}(S)$.