The isomorphism problem for toral relatively hyperbolic groups
François Dahmani and Daniel Groves

The isomorphism problem for torsion free relatively hyperbolic groups with abelian parabolic subgroups is decidable. Special cases of this statement include Limit groups (see also work of Bumagin, Kharlampovich, and Miasnikov) and torsion free hyperbolic groups. For this latter case, Sela proposed a solution for groups with no essencial splitting, in 1995, and has an unpublished proof of the general case. We propose a method inspired by his ideas, but simplified and improved. A first important tool is the decidability of existential theory for torsion free (relatively) hyperbolic groups, and some refinements. This allows to list all conjugacy classes of monomorphisms between two such groups, provided there are only finitely many. The second important tool is the computability of the essential JSJ decomposition of such groups.



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