Reducing the conjugacy problem for relatively hyperbolic automorphisms to peripheral components

TL;DR

This paper presents a reduction method for the conjugacy problem in outer automorphisms of free and hyperbolic groups, linking it to algorithmic problems on mapping tori, and solves it for new automorphism classes.

Contribution

It introduces a novel reduction approach that simplifies the conjugacy problem to manageable algorithmic problems on peripheral components.

Findings

Solved conjugacy problem for several new classes of automorphisms.

Proposed a pathway toward a complete solution for $Out(F_n)$.

Linked conjugacy problem to algorithmic problems on mapping tori.

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 $Out (F_n)$.