Equidistribution and counting for orbits of geometrically finite hyperbolic groups

By H. Oh, Nimish A. Shah. 2010

96 citations recorded by the source corpus.

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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/fa40cb71775d27078309c0fcdafbce79794a90e3

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Citations in the corpus

Apollonian circle packings and closed horospheres on hyperbolic 3-manifolds (2008, result): “See also [32] for similar counting results for hyperbolic and spherical Apollonian circle packings.”
Notes on Thin Matrix Groups (2012, methodology): “Thanks to the referee for pointing me to the relevant applications in [Gromov and Guth 2012].”
Matrix coefficients, counting and primes for orbits of geometrically finite groups (2012, methodology): “14]; however, as opposed to [52], we are here working in 0\G rather than in T1(0\Hn).”
Closed geodesics and holonomies for Kleinian manifolds (2014, methodology): “([18], [14], [24]) For any functions Ψ1,Ψ2 ∈ Cc(Γ\G),”
Anosov groups: local mixing, counting and equidistribution (2020, methodology): “13) When G has rank one, this is proved in [27] and [23] for any geometrically finite groups Γ.”