Amenable groups and stabilizers of measures on the boundary of a Hadamard manifold

Abstract

Let X be a symmetric space of noncompact type or more generally a Hadamard manifold, i.e. a complete simply connected Riemannian manifold of nonpositive sectional curvature. We consider a finite positive Borel-measure # on the ideal boundary X(~) (see Sect. 2 for the definitions). Let G be the isometry group of X and G~ the subgroup which stabilizes the measure #. In the case of a symmetric space we obtain the following result.