Hyperbolic manifolds with geodesic boundary which are determined by their fundamental group

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.

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