Equidistribution and Counting for orbits of geometrically finite hyperbolic groups

TL;DR

This paper develops an asymptotic counting formula for orbits of geometrically finite hyperbolic groups acting on vector spaces, linking ergodic theory, geometric finiteness, and orbit distribution.

Contribution

It provides a new asymptotic formula for orbit counting in hyperbolic groups with conditions on measures and skinning size, extending previous results to more general settings.

Findings

Derived an asymptotic count for orbit points of bounded norm.

Established conditions for finiteness of the skinning size in geometrically finite groups.

Connected orbit distribution with ergodic properties of hyperbolic manifolds.

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.