Equidistribution and counting for orbits of geometrically finite hyperbolic groups

1.1. Motivation and overview. Let G denote the identity component of the special orthogonal group SO(n, 1), n ≥ 2, and V a finite-dimensional real vector space on which G acts linearly from the right. A discrete subgroup of a locally compact group with finite covolume is called a lattice. For v ∈ V and a subgroup H of G, let Hv = {h ∈ H : vh = v} denote the stabilizer of v in H. A subgroup H of G is called symmetric if there exists a non-trivial involutive automorphism σ of G such that the identity component of H is the same as the identity component of G = {g ∈ G : σ(g) = g}. Theorem 1.1 (Duke-Rudnick-Sarnak [9]). Fix w0 ∈ V such that Gw0 is symmetric. Let Γ be a lattice in G such that Γw0 is a lattice in Gw0 . Then for any norm ‖·‖ on V ,