Journal of the American Mathematical Society

Abstract:

Let $G$ be the identity component of $\mathrm {SO}(n,1)$, $n\ge 2$, 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 $\mathbb {R} w_0$ is a parabolic subgroup of $G$. For any non-elementary discrete subgroup $\Gamma$ of $G$ with its orbit $w_0\Gamma$ discrete, we compute an asymptotic formula (as $T\to \infty$) for the number of points in $w_0\Gamma$ of norm at most $T$, provided that the Bowen-Margulis-Sullivan measure on $\mathrm {T}^1(\Gamma \backslash \mathbb {H}^n)$ 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 $\Gamma \backslash \mathbb {H}^n$. We also give a criterion on the finiteness of the $\Gamma$-skinning size of $w_0$ for $\Gamma$ geometrically finite.