  We solve Dehn's isomorphism problem for virtually torsion-free relatively
hyperbolic groups with nilpotent parabolic subgroups.
  We do so by reducing the isomorphism problem to three algorithmic problems in
the parabolic subgroups, namely the isomorphism problem, separation of torsion
(in their outer automorphism groups) by congruences, and the mixed Whitehead
problem, an automorphism group orbit problem. The first step of the reduction
is to compute canonical JSJ decompositions. Dehn fillings and the given
solutions of the algorithmic problems in the parabolic groups are then used to
decide if the graphs of groups have isomorphic vertex groups and, if so,
whether a global isomorphism can be assembled.
  For the class of finitely generated nilpotent groups, we give solutions to
these algorithmic problems by using the arithmetic nature of these groups and
of their automorphism groups.
