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

### Michael Brandenbursky

Ben Gurion University of the Negev, Beer Sheva, Israel### Michał Marcinkowski

Ben Gurion University of the Negev, Beer Sheva, Israel, Regensburg University, Germany, and Wroclaw University, Poland

## Abstract

Let $\mathbf{F}_n$ be the free group on $n$ generators and $\boldsymbol{\Gamma}_g$ 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 $\boldsymbol{\Gamma}_g$. 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.

## Cite this article

Michael Brandenbursky, Michał Marcinkowski, Aut-invariant norms and Aut-invariant quasimorphisms on free and surface groups. Comment. Math. Helv. 94 (2019), no. 4, pp. 661–687

DOI 10.4171/CMH/470