The isomorphism problem for all hyperbolic groups

TL;DR

This paper provides a comprehensive algorithmic solution to the isomorphism problem for all hyperbolic groups, including those with torsion, and addresses related automorphism and orbit problems.

Contribution

It introduces a complete solution to Dehn's isomorphism problem for hyperbolic groups and extends it to groups with peripheral structures, also solving Whitehead's problem.

Findings

Algorithm for isomorphism problem for hyperbolic groups

Solution to Whitehead's problem in hyperbolic groups

Computes automorphism groups preserving peripheral structures

Abstract

We give a solution to Dehn's isomorphism problem for the class of all hyperbolic groups, possibly with torsion. We also prove a relative version for groups with peripheral structures. As a corollary, we give a uniform solution to Whitehead's problem asking whether two tuples of elements of a hyperbolic group $G$ are in the same orbit under the action of $\Aut(G)$. We also get an algorithm computing a generating set of the group of automorphisms of a hyperbolic group preserving a peripheral structure.