(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 $Out(F_{n})$.[-]