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

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 $\mathrm{SAut}({\mathrm{F}}_{\mathrm{n}})$ 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 $2$.

42 pages, 1 figure; v2: Final version, to appear in Proc. Lond. Math. Soc

References in corpus (1)

• Outer actions of $\mathrm{Out}(F_n)$ on small right-angled Artin groups

Cited by in corpus (2)

• Generating Sets and Algebraic Properties of Pure Mapping Class Groups of Infinite Graphs
• Problems on handlebody groups