# Groups, Geometry, and Dynamics

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**Volume 5, Issue 1, 2011, pp. 39–106**

**DOI: 10.4171/GGD/116**

Published online: 2011-01-19

The Recognition Theorem for Out(`F`_{n})

^{[1]}and Michael Handel

^{[2]}(1) Rutgers University, Newark, USA

(2) Lehman College, CUNY, Bronx, USA

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.

*Keywords: *Outer automorphisms, free group

Feighn Mark, Handel Michael: The Recognition Theorem for Out(`F`_{n}). *Groups Geom. Dyn.* 5 (2011), 39-106. doi: 10.4171/GGD/116