Reducing the conjugacy problem for relatively hyperbolic automorphisms to peripheral components

Abstract

We give a reduction of the conjugacy problem among outer automorphisms of free (and torsion-free hyperbolic) groups to specific algorithmic problems pertaining to mapping tori of polynomially growing automorphisms. We explain how to use this reduction and solve the conjugacy problem for several new classes of outer automorphisms. This proposes a path toward a full solution to the conjugacy problem for $Out (F_n)$.