Equidistribution and Counting for orbits of geometrically finite hyperbolic groups

Details Equidistribution and Counting for orbits of geometrically finite hyperbolic groups Equidistribution and Counting for orbits of geometrically finite hyperbolic groups

2010-01-13

arxiv.org/abs/1001.2096v4

Mathematics

Dynamical Systems

Main counting results are generalized to all symmetric homogeneous varieties of G and the equidistribution results are obtaine

Scientific paper

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 is 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.

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