Translation lengths of outer automorphisms of finitely generated free-by-finite groups

TL;DR

This paper extends the understanding of outer automorphism groups from free groups to free-by-finite groups, showing similar subgroup dichotomies and translation properties.

Contribution

It generalizes known results about $ extrm{Out}(F_n)$ to the broader class of free-by-finite groups, establishing translation discreteness and subgroup structure.

Findings

$ extrm{Out}(G)$ is translation discrete

Subgroups of $ extrm{Out}(G)$ are either virtually abelian or contain a free group

Generalizes subgroup dichotomy from free groups to free-by-finite groups

Abstract

Bestvina, Feighn and Handel proved that every subgroup of the outer automorphism group, $\textrm{Out}(F_n)$, of the free group of rank $n$ is either virtually finitely generated abelian or contains a nonabelian free group. In this note we consider the more general situation of the outer automorphism group $\textrm{Out}(G)$ of a finitely generated free-by-finite group $G$. We show that $\textrm{Out}(G)$ is translation discrete and that every subgroup of $\textrm{Out}(G)$ is either virtually finitely generated abelian or contains a nonabelian free group.