Maximal volume representations are fuchsian

We prove a volume-rigidity theorem for Fuchsian representations of fundamental groups of hyperbolic k-manifolds into Isom(ℍn), Namely, we show that if M is a complete hyperbolic k-manifold with finite volume, then the volume of any representation of π1(M) into Isom(ℍn), 3≤k≤n, is less than the volume of M, and the volume is maximal if and only if the representation is discrete, faithful and 'k-Fuchsian'. © Springer 2006.