# On the smallest non-abelian quotient of $\mathrm{Aut}(F_n)$

## TL;DR

This paper proves that the smallest non-abelian quotient of the automorphism group of a free group is a specific projective special linear group over the field with two elements, confirming a prior conjecture and providing new bounds and representation theory results.

## Contribution

It confirms the conjecture that the smallest non-abelian quotient of Aut(F_n) is PSL_n(Z/2Z) and introduces exponential lower bounds and new representation theory insights for SAut(F_n).

## Key findings

- The smallest non-abelian quotient of Aut(F_n) is PSL_n(Z/2Z).
- An exponential lower bound for the size of sets where SAut(F_n) acts non-trivially.
- New results on the representation theory of SAut(F_n) in small dimensions and characteristics.

## Abstract

We show that the smallest non-abelian quotient of $\mathrm{Aut}(F_n)$ is $\mathrm{PSL}_n(\mathbb{Z}/2\mathbb{Z}) = \mathrm{L}_n(2)$, thus confirming a conjecture of Mecchia--Zimmermann. In the course of the proof we give an exponential (in $n$) lower bound for the cardinality of a set on which $\mathrm{SAut}(F_n)$, the unique index $2$ subgroup of $\mathrm{Aut}(F_n)$, can act non-trivially. We also offer new results on the representation theory of $\mathrm{SAut(F_n)}$ in small dimensions over small, positive characteristics, and on rigidity of maps from $\mathrm{SAut}(F_n)$ to finite groups of Lie type and algebraic groups in characteristic $2$.