Random walks and contracting elements III: Outer space and outer automorphism group

TL;DR

This paper explores random walks on metric spaces with the BGIP property, revealing that generic outer automorphisms of free groups have distinct forward and backward expansion factors, answering a question by Handel and Mosher.

Contribution

It extends the study of random walks to asymmetric spaces and demonstrates a novel property of outer automorphisms regarding their expansion factors.

Findings

Generic outer automorphisms have different forward and backward expansion factors

The study applies the BGIP property to analyze isometries in metric spaces

Answers a previously open question by Handel and Mosher

Abstract

Continuing from the author's previous article 'Random walks and contracting elements I', we study random walks on (possibly asymmetric) metric spaces using the bounded geodesic image property (BGIP) of certain isometries. As an application, we show that a generic outer automorphism of the free group of rank at least 3 has different forward and backward expansion factors. This answers a question of Handel and Mosher.