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

Abstract

For any finitely generated group $G$, two complexity functions $α_G$ and $β_G$ 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 $α_2$ and $β_2$ is computed. For rank $r\geqslant 3$, polynomial lower bounds are provided for $α_r$ and $β_r$, and the existence of a polynomial upper bound is proved for $β_r$.