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

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 $r⩾3$, polynomial lower bounds are provided for ${\alpha }_r$ and ${\beta }_r$, and the existence of a polynomial upper bound is proved for ${\beta }_r$.