On the distribution of orbits of geometrically finite hyperbolic groups on the boundary

Abstract. We investigate the distribution of orbits of a non-elementary dis-crete hyperbolic subgroup Γ acting on Hn and its geometric boundary ∂∞(Hn). In particular, we show that if Γ admits a finite Bowen-Margulis-Sullivan mea-sure (for instance, if Γ is geometrically finite), then every Γ-orbit in ∂∞(Hn) is equidistributed with respect to the Patterson-Sullivan measure supported on the limit set Λ(Γ). The appendix by Maucourant is the extension of a part of his thesis where he obtains the same result as a simple application of Roblin’s theorem. Our approach is via establishing the equidistribution of solvable flows on the unit tangent bundle of Γ\Hn, which is of independent interest. 1