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

Abstract

Bestvina, Feighn and Handel proved that every subgroup of the outer automorphism group, $\text{Out}(F_n)$, 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 $\text{Out}(G)$ of a finitely generated free-by-finite group $G$. We show that $\text{Out}(G)$ is translation discrete and that every subgroup of $\text{Out}(G)$ is either virtually finitely generated abelian or contains a nonabelian free group.

8 pages. v2: incorporates changes suggested by the referee, improved Corollary 3.7, added references