Stability in Outer Space

TL;DR

This paper characterizes strongly Morse quasi-geodesics in Outer space and defines convex cocompact subgroups of Out(F_n) through their actions, drawing parallels with mapping class groups.

Contribution

It introduces a new characterization of strongly Morse quasi-geodesics and defines convex cocompact subgroups of Out(F_n) based on their actions in Outer space.

Findings

Strongly Morse quasi-geodesics project to quasi-geodesics in the free factor graph.

Convex cocompact subgroups are characterized by their orbit maps being quasi-isometric embeddings.

The approach parallels the characterization of convex cocompact subgroups in mapping class groups.

Abstract

We characterize strongly Morse quasi-geodesics in Outer space as quasi-geodesics which project to quasi-geodesics in the free factor graph. We define convex cocompact subgroups of $Out(F_n)$ as subgroups such that an orbit map in the free factor graph is a quasi-isometric embedding, and we characterize such groups via their action on Outer space in a way which resembles the characterization of convex cocompact subgroups of mapping class groups.