  Let M and N be n-dimensional connected orientable finite-volume hyperbolic
manifolds with geodesic boundary, and let f be a given isomorphism between the
fundamental groups of M and N. We study the problem whether there exists an
isometry between M and N which induces f.
  We show that this is always the case if the dimension of M and N is at least
four, while in the three-dimensional case the existence of an isometry inducing
f is proved under some (necessary) additional conditions on f. Such conditions
are trivially satisfied if the boundaries of M and N are both compact.
