Hyperbolic volume of representations of fundamental groups of cusped 3-manifolds.

Let W be a compact manifold and let R be a representation of its fundamental group into PSL(2,C). Then the volume of R is defined by taking any R-equivariant map from the universal cover of W to H^3 and then by integrating the pull-back of the hyperbolic volume form on a fundamental domain. It turns out that such a volume does not depend on the choice of the equivariant map. Dunfield extended this construction to the case of a non-compact (cusped) manifold M, but he did not prove the volume is well-defined in all cases. We prove here that the volume of a representation is always welldefined and depends only on the representation. Moreover, we show that this volume can be easily computed by straightening any ideal triangulation of M. We show that the volume of a representation is bounded from above by the relative simplicial volume of M. Finally, we prove a rigidity theorem for representations of the fundamental group of a hyperbolic manifold. Namely, we prove that if M is hyperbolic and vol(R) = vol(M) then R is discrete and faithful.