Equidistribution and Counting for orbits of geometrically finite hyperbolic groups

Abstract

Let G be the identity component of SO(n,1), acting linearly on a finite dimensional real vector space V. Consider a vector w_0 in V such that the stabilizer of w_0 is a symmetric subgroup of G or the stabilizer of the line Rw_0 is a parabolic subgroup of G. For any non-elementary discrete subgroup Gamma of G with w_0Gamma discrete, we compute an asymptotic formula for the number of points in w_0Gamma of norm at most T, provided that the Bowen-Margulis-Sullivan measure on the associated hyperbolic manifold and the Gamma skinning size of w_0 are finite. The main ergodic ingredient in our approach is the description for the limiting distribution of the orthogonal translates of a totally geodesically immersed closed submanifold of Gamma\H^n. We also give a criterion on the finiteness of the Gamma skinning size of w_0 for Gamma geometrically finite.

Extensions of Equidistribution results to G/Gamma are obtained for Gamma Zariski dense, and Much more precise description on the structure of cuspidal neighborhoods of parabolic points is obtained for Gamma geometrically finite. 63 pages (with 1 figure)

References in corpus (3)

• The Asymptotic distribution of circles in the orbits of Kleinian groups
• Counting visible circles on the sphere and Kleinian groups
• Limits of translates of divergent geodesics and Integral points on one-sheeted hyperboloids

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• Exponential mixing of frame flows for convex cocompact hyperbolic manifolds
• Classification of joinings for Kleinian groups
• Orbit closures of unipotent flows for hyperbolic manifolds with Fuchsian ends
• Exponential mixing of frame flows for geometrically finite hyperbolic manifolds
• Exponential mixing of geodesic flows for geometrically finite hyperbolic manifolds with cusps
• Directions in Orbits of Geometrically Finite Hyperbolic Subgroups
• Joint partial equidistribution of Farey rays in negatively curved manifolds and trees
• Generalization of Selberg's $3/16$ theorem for convex cocompact thin subgroups of $\mathrm{SO}(n,1)$