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

## TL;DR

This paper characterizes distorted elements in Aut-invariant word metrics on free and surface groups, reestablishes Stallings theorem, answers Calegari's growth question, and constructs infinitely many Aut-invariant quasimorphisms on free groups.

## Contribution

It provides a complete characterization of distorted elements, reproof of Stallings theorem, and constructs new Aut-invariant quasimorphisms, addressing open problems in the field.

## Key findings

- Complete characterization of distorted and undistorted elements.
- Reproof of Stallings theorem.
- Construction of infinitely many Aut-invariant quasimorphisms.

## 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 Mikl\'os Ab\'ert.