Normal closure of finite subgroups of $\mathrm{Aut}(F_n)$ and $\mathrm{Out}(F_n)$

TL;DR

This paper characterizes the normal closures of finite subgroups in automorphism and outer automorphism groups of free groups, showing they are typically the special automorphism groups unless the subgroup is contained within them, with partial results for 2-power order groups.

Contribution

It provides a classification of the normal closures of finite subgroups in $ ext{Aut}(F_n)$ and $ ext{Out}(F_n)$, extending understanding of subgroup structure in these automorphism groups.

Findings

Normal closure of G is $ ext{SAut}(F_n)$ if G is in $ ext{SAut}(F_n)$ and |G| not a power of 2.

Normal closure of G' is $ ext{SOut}(F_n)$ if G' is in $ ext{SOut}(F_n)$ and |G'| not a power of 2.

Partial theorems for groups of order a power of 2.

Abstract

For $n\geq 3$, let $G$ be a nontrivial finite subgroup of $\mathrm{Aut}(F_n)$ with $|G|$ not a power of $2$. We prove that the normal closure $N(G)$ is $\mathrm{SAut}(F_n)$ if $G\subset\mathrm{SAut}(F_n)$ and $N(G)$ is $\mathrm{Aut}(F_n)$ otherwise. When $|G|$ is a power of $2$, we have a partial theorem. Similarly, let $G'$ be a nontrivial finite subgroup of $\mathrm{Out}(F_n)$ with $|G'|$ not a power of $2$. Then the normal closure $N(G')$ is $\mathrm{SOut}(F_n)$ if $G'\subset\mathrm{SOut}(F_n)$ and $N(G')$ is $\mathrm{Out}(F_n)$ otherwise. When $|G'|$ is a power of $2$, we have a partial theorem as well.