The conjugacy problem for Out(F3)

We present a solution to the conjugacy problem in the group of outer automorphisms of F3, a free group of rank 3. We distinguish according to several computable invariants such as irreducibility, subgroups of polynomial growth, and subgroups carrying the attracting lamination. We establish by considerations on train tracks that the conjugacy problem is decidable for the outer automorphisms of F3 that preserve a given rank 2 free factor. Then we establish by consideration on mapping tori that it is decidable for outer automorphisms of F3 whose maximal polynomial growth subgroups are cyclic. This covers all the cases left by the state of the art.