Distortion for Abelian Subgroups of $\mathrm{Out}(F_n)$

TL;DR

This paper proves that abelian subgroups of the outer automorphism group of a free group are quasi-isometrically embedded and confirms the rank conjecture for this group, using train track map theory.

Contribution

It establishes the quasi-isometric embedding of abelian subgroups and proves the rank conjecture for Out(F_n), advancing understanding of its subgroup structure.

Findings

Abelian subgroups are quasi-isometrically embedded in Out(F_n)

The rank conjecture for Out(F_n) is proven

Utilizes train track map theory by Feighn-Handel

Abstract

We prove that abelian subgroups of the outer automorphism group of a free group are quasi-isometrically embedded. Our proof uses recent developments in the theory of train track maps by Feighn-Handel. As an application, we prove the rank conjecture for $\mathrm{Out}(F_n)$.