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

Abstract

Let $F_n$ be the free group on $n$ generators and ${\Gamma }_g$ the surface group of genus $g$. We consider two particular generating sets: the set of all primitive elements in $F_n$ and the set of all simple loops in ${\Gamma }_g$. 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 $F_2$ that are $Aut(F_2)$-invariant. This answers an open problem posed by Miklós Abért.

arXiv comment26 pages, 3 figures. Accepted in Commentarii Mathematici Helvetici