Résumé : (joint work with Michael Handel) $Out(F_n):=Aut(F_n)/Inn(F_n)$ denotes the outer automorphism group of the rank n free group $F_n$. An element $f$ of $Out(F_n)$ is polynomially growing if the word lengths of conjugacy classes in Fn grow at most polynomially under iteration by $f$. The existence in $Out(F_n),n>2$, of elements with non-linear polynomial growth is a feature of $Out(F_n)$ not shared by mapping class groups of surfaces.
To avoid some finite order behavior, we restrict attention to the subset $UPG(F_n)$ of $Out(F_n)$ consisting of polynomially growing elements whose action on $H_1(F_n,Z)$ is unipotent. In particular, if $f$ is polynomially growing and acts trivially on $H_1(F_n,Z_3)$ then $f$ is in $UPG(F_n)$ and further every polynomially growing element of $Out(F_n)$ has a power that is in $UPG(F_n)$. The goal of the talk is to describe an algorithm to decide given $f,g$ in $UPG(F_n)$ whether or not there is h in $Out(F_n)$ such that $hf{h}^{-1}=g$.
The conjugacy problem for linearly growing elements of $UPG(F_n)$ was solved by Cohen-Lustig. Krstic-Lustig-Vogtmann solved the case of linearly growing elements of $Out(F_n)$.
A key technique is our use of train track representatives for elements of $Out(F_n)$, a method pioneered by Bestvina-Handel in the early 1990s that has since been ubiquitous in the study of $Out(F_n)$.

Keywords : Out(Fn); conjugacy problem; train tracks