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

TL;DR

This paper investigates the dynamical behavior of subgroups within the outer automorphism group of a free group, establishing the existence of elements with polynomial growth properties that characterize the subgroup’s overall dynamics.

Contribution

It proves that for any subgroup of f(n), there exists an element with polynomial growth behavior that reflects the subgroup's dynamical properties.

Findings

Existence of elements with polynomial growth in subgroups of f(n)

Characterization of subgroup dynamics via polynomial growth elements

Extension of dynamical properties to all elements in a subgroup

Abstract

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