Strongly Contracting Geodesics in Outer Space

TL;DR

This paper investigates the geometric properties of Outer Space under the Lipschitz metric, showing that certain automorphisms act hyperbolically with strongly contracting axes, leading to stability results for quasi-geodesics.

Contribution

It establishes that fully irreducible elements of Out(F_n) have strongly contracting axes in Outer Space, providing new insights into their geometric dynamics.

Findings

Fully irreducible automorphisms act by hyperbolic isometries

Axes of these automorphisms are strongly contracting

Quasi-geodesics with endpoints on axes are stable

Abstract

We study the Lipschitz metric on Outer Space and prove that fully irreducible elements of Out(F_n) act by hyperbolic isometries with axes which are strongly contracting. As a corollary, we prove that the axes of fully irreducible automorphisms in the Cayley graph of Out(F_n) are stable, meaning that a quasi-geodesic with endpoints on the axis stays within a bounded distance from the axis.