Contracting orbits in Outer space

TL;DR

This paper explores the properties of strongly contracting geodesics in Outer space, establishing their relationship with quasigeodesics in the free factor complex, and provides new examples and criteria for subgroup actions.

Contribution

It proves a converse to Bestvina--Feighn's theorem, linking contracting geodesics in Outer space to quasigeodesics in the free factor complex, and introduces new examples and subgroup criteria.

Findings

Strongly contracting geodesics project to quasigeodesics in the free factor complex

Provides a converse to a theorem of Bestvina--Feighn

Constructs new examples of strongly contracting geodesics in Outer space

Abstract

We show that strongly contracting geodesics in Outer space project to parameterized quasigeodesics in the free factor complex. This result provides a converse to a theorem of Bestvina--Feighn, and is used to give conditions for when a subgroup of $\mathrm{Out}(\mathbb{F})$ has a quasi-isometric orbit map into the free factor complex. It also allows one to construct many new examples of strongly contracting geodesics in Outer space.