Polynomial growth and subgroups of $\mathrm{Out}(F_{\mathtt{n}})$

Abstract

This paper, which is the last of a series of three papers, studies dynamical properties of elements of $\mathrm{Out}(F_{\mathtt{n}})$, the outer automorphism group of a nonabelian free group $F_{\mathtt{n}}$. We prove that, for every subgroup $H$ of $\mathrm{Out}(F_{\mathtt{n}})$, there exists an element $\varphi \in H$ such that, for every element $g$ of $F_{\mathtt{n}}$, the conjugacy class $[g]$ has polynomial growth under iteration of $\varphi$ if and only if $[g]$ has polynomial growth under iteration of every element of $H$.

31 pages