On the Distribution of Orbits of Geometrically Finite Hyperbolic Groups on the Boundary (with Appendix by François Maucourant)

We investigate the distribution of orbits of a non-elementary discrete hyperbolic subgroup Γ acting on Hn and its geometric boundary ∂∞(H). In particular, we show that if Γ admits a finite Bowen-Margulis-Sullivan measure (for instance, if Γ is geometrically finite), then every Γ-orbit in ∂∞(H) 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.