ON THE RIGIDITY OF DISCRETE ISOMETRY GROUPS OF NEGATIVELY CURVED SPACES

We prove an ergodic rigidity theorem for discrete isometry groups of C AT(-1) spaces. We give explicit examples of divergence isometry groups with infinite covolume in the case of trees, piecewise hyperbolic 2-p olyhedra, hyperbolic Bruhat-Tits buildings and rank one symmetric spac es. We prove that two negatively curved Riemannian metrics, with conic al singularities of angles st least 2 pi, on a closed surface, with bo undary map absolutely continuous with respect to the Patterson-Sulliva n measures, are isometric. For that, we generalize J.-P. Otal's result to prove that a negatively curved Riemannian metric, with conical sin gularities of angles at least 2 pi, on a closed surface, is determined , up to isometry, by its marked length spectrum.