# The Recognition Theorem for Out(<var>F</var><sub><var>n</var></sub>)

### Mark Feighn

Rutgers University, Newark, USA### Michael Handel

Lehman College, CUNY, Bronx, USA

## 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 $\phi\in\mathrm{Out}(F_n)$, $\phi^{K_n}$ is forward rotationless. An important part of our analysis is to show that rotationless elements are represented by particularly nice relative train track maps.

## Cite this article

Mark Feighn, Michael Handel, The Recognition Theorem for Out(<var>F</var><sub><var>n</var></sub>). Groups Geom. Dyn. 5 (2011), no. 1, pp. 39–106

DOI 10.4171/GGD/116