Aut-invariant norms and Aut-invariant quasimorphisms on free and surface groups

Abstract

Let ${\mathbf{F}}_n$ be the free group on $n$ generators and ${\mathbf{\Gamma }}_g$ the surface group of genus $g$. We consider two particular generating sets: the set of all primitive elements in ${\mathbf{F}}_n$ and the set of all simple loops in ${\mathbf{\Gamma }}_g$. We give a complete characterization of distorted and undistorted elements in the corresponding Aut-invariant word metrics. In particular, we reprove Stallings theorem and answer a question of Danny Calegari about the growth of simple loops. In addition, we construct infinitely many quasimorphisms on ${\mathbf{F}}_2$ that are Aut$({\mathbf{F}}_2)$-invariant. This answers an open problem posed by Miklós Abért.