Equidistribution and counting for orbits of geometrically finite hyperbolic groups

Abstract

Let G G be the identity component of S O ( n , 1 ) \mathrm {SO}(n,1) , n ≥ 2 n\ge 2 , acting linearly on a finite-dimensional real vector space V V . Consider a vector w 0 ∈ V w_0\in V such that the stabilizer of w 0 w_0 is a symmetric subgroup of G G or the stabilizer of the line R w 0 \mathbb {R} w_0 is a parabolic subgroup of G G . For any non-elementary discrete subgroup Γ \Gamma of G G with its orbit w 0 Γ w_0\Gamma discrete, we compute an asymptotic formula (as T → ∞ T\to \infty ) for the number of points in w 0 Γ w_0\Gamma of norm at most T T , provided that the Bowen-Margulis-Sullivan measure on T 1 ( Γ ∖ H n ) \mathrm {T}^1(\Gamma \backslash \mathbb {H}^n) and the Γ \Gamma -skinning size of w 0 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 Γ ∖ H n \Gamma \backslash \mathbb {H}^n . We also give a criterion on the finiteness of the Γ \Gamma -skinning size of w 0 w_0 for Γ \Gamma geometrically finite.