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

Let M 1 and M 2 be n-dimensional connected orientable finite-volume hyperbolic manifolds with geodesic boundary, and let φ : π 1 ( M 1 ) → π 1 ( M 2 ) be a given group isomorphism. We study the problem whether there exists an isometry ψ : M 1 → M 2 such that ψ * = φ . We show that this is always the case if n ⩾ 4 , while in the 3-dimensional case the existence of ψ is proved under some (necessary) additional conditions on φ. Such conditions are trivially satisfied if ∂ M 1 and ∂ M 2 are both compact.