There is a paper of Martin Lustig on his webpage giving a positive answer to the conjugacy problem for the outer automorphism group of the free group $F\_n$. On the other hand, there seems not to be a publication in a journal about this, but several related publications. Is it possible to say which are the "best" results in this direction, with a complete proof ? (For example, a result by Sela if the outer automorphism group is finite, etc.).



Is the conjugacy problem solvable in $Out(F_n)$?
One result in this direction is given by [Dahmani](https://arxiv.org/abs/1307.2108) . His algorithm will determine conjugacy for pairs of *atoroidal* outer automorphisms, ie automorphisms that do not fix a non-trivial conjugacy class. Note that he uses some very heavy machinery---his solution goes via the isomorphism problem for hyperbolic groups (solved by him and Guirardel along the lines pioneered by Sela).


