Bounding the gap between a free group (outer) automorphism and its inverse

TL;DR

This paper investigates the maximal differences in complexity between automorphisms of free groups and their inverses, providing exact asymptotics for rank 2 and bounds for higher ranks.

Contribution

It introduces complexity functions for automorphisms of finitely generated groups and determines their asymptotic behavior for free groups of various ranks.

Findings

Exact asymptotics for rank 2 free groups.

Polynomial lower bounds for ranks ≥ 3.

Existence of polynomial upper bounds for higher ranks.

Abstract

For any finitely generated group $G$, two complexity functions $\alpha_G$ and $\beta_G$ are defined to measure the maximal possible gap between the norm of an automorphism (respectively outer automorphism) of $G$ and the norm of its inverse. Restricting attention to free groups $F_r$, the exact asymptotic behaviour of $\alpha_2$ and $\beta_2$ is computed. For rank $r\geqslant 3$, polynomial lower bounds are provided for $\alpha_r$ and $\beta_r$, and the existence of a polynomial upper bound is proved for $\beta_r$.