The isomorphism problem for all hyperbolic groups

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.

71 pages, 4 figures

References in corpus (4)

• Foliations for solving equations in groups: free, virtually free, and hyperbolic groups
• On the isomorphism problem for generalized Baumslag-Solitar groups
• The Isomorphism Problem for Toral Relatively Hyperbolic Groups
• Detecting free splittings in relatively hyperbolic groups

Cited by in corpus (13)

• Tadpole Labelled Oriented Graph Groups and Cyclically Presented Groups
• Recognizing a relatively hyperbolic group by its Dehn fillings
• McCool groups of toral relatively hyperbolic groups
• Deciding Isomorphy using Dehn fillings, the splitting case
• Computing JSJ decompositions of hyperbolic groups
• On suspensions, and conjugacy of hyperbolic automorphisms (and of a few more)
• On automorphisms and splittings of special groups
• Regularity of quasigeodesics characterises hyperbolicity
• Reducing the conjugacy problem for relatively hyperbolic automorphisms to peripheral components
• The JSJ-decompositions of one-relator groups with torsion
• Immersed cycles and the JSJ decomposition
• JSJ decompositions of doubles of free groups
• The Outer Automorphism Groups of Two-Generator One-Relator Groups with Torsion