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

Abstract

AbstractWe investigate the distribution of orbits of a non-elementary discrete hyperbolic subgroup Γ acting on ℍn and its geometric boundary ∂∞(ℍn). 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 ∂∞(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 PhD 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.