On suspensions, and conjugacy of hyperbolic automorphisms (and of a few more)

Abstract

Part 1 : We remark that the conjugacy problem for pairs of hyperbolic au- tomorphisms of a finitely presented group (typically a free group) is decidable. The solution that we propose uses the isomorphism problem for the suspensions, and the study of their automorphism group. Part 2 : In a previous work (part 1), we remarked that the conjugacy problem for pairs of atoroidal automorphisms of a free group was solvable by mean of the isomorphism problem for hyperbolic groups and an orbit problem for the automorphism group of their suspensions (i.e. their semidirect product with Z for the relevant structural automorphism). We consider the same problem a few more automorphisms of free groups, those that produce relatively hyperbolic suspensions that do not split over a parabolic subgroup.