The Isomorphism Problem for All Hyperbolic Groups

François Dahmani (1) , Vincent Guirardel (2)

1 IF - Institut Fourier
2 IRMAR - Institut de Recherche Mathématique de Rennes

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.

Mots clés: Hyperbolic groups isomorphism problem graph of groups JSJ decompositions orbifolds

Domaines: Théorie des groupes [math.GR]

François Dahmani, Vincent Guirardel. The Isomorphism Problem for All Hyperbolic Groups. Geometric And Functional Analysis, 2011, 21 (2), pp.20F67 (20F10 20F65 57M50 20E06 20E08). ⟨10.1007/s00039-011-0120-0⟩. ⟨hal-01101751⟩