Pulling back stability with applications to Out($F_n$) and relatively hyperbolic groups

TL;DR

This paper demonstrates that stability, a key quasiconvexity property, is preserved under proper actions on metric spaces, with applications to groups like Out($F_n$) and relatively hyperbolic groups.

Contribution

It establishes the pullback of stability under proper actions and characterizes stability in certain relatively hyperbolic groups, extending understanding of subgroup properties.

Findings

Stability pulls back under proper actions on metric spaces.

Convex cocompact subgroups in mapping class groups and Out($F_n$) are stable.

Stability characterized in relatively hyperbolic groups with linear divergence.

Abstract

We prove that stability -- a strong quasiconvexity property -- pulls back under proper actions on proper metric spaces. This result has several applications, including that convex cocompact subgroups of both mapping class groups and outer automorphism groups of free groups are stable. We also characterize stability in relatively hyperbolic groups whose parabolic subgroups have linear divergence.