The Recognition Theorem for $\mathrm{Out}(F_n)$

Abstract

Our goal is to find dynamic invariants that completely determine elements of the outer automorphism group $\mathrm{Out}(F_n)$ of the free group $F_n$ of rank $n$. To avoid finite order phenomena, we do this for forward rotationless elements. This is not a serious restriction. For example, there is $K_n>0$ depending only on $n$ such that, for all $\varphi \in \mathrm{Out}(F_n)$, ${\varphi }^{K_n}$ is forward rotationless. An important part of our analysis is to show that rotationless elements are represented by particularly nice relative train track maps.