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

TL;DR

This paper demonstrates that the conjugacy problem for pairs of hyperbolic automorphisms of finitely presented groups, especially free groups, is decidable by analyzing their suspensions and automorphism groups, extending previous results to more automorphisms.

Contribution

It extends the decidability of the conjugacy problem to a broader class of automorphisms of free groups using suspension isomorphism techniques.

Findings

Conjugacy problem for hyperbolic automorphisms is decidable.

Method applies to automorphisms producing relatively hyperbolic suspensions.

Extends previous results to new classes of automorphisms.

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.