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Equidistribution and counting for orbits of geometrically finite hyperbolic groups


Authors: Hee Oh and Nimish A. Shah
Journal: J. Amer. Math. Soc. 26 (2013), 511-562
MSC (2010): Primary 11N45, 37F35, 22E40; Secondary 37A17, 20F67
DOI: https://doi.org/10.1090/S0894-0347-2012-00749-8
Published electronically: October 2, 2012
MathSciNet review: 3011420
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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.


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Additional Information

Hee Oh
Affiliation: Department of Mathematics, Brown University, Providence, Rhode Island 02912 and Korea Institute for Advanced Study, Seoul, Korea
MR Author ID: 615083
Email: heeoh@math.brown.edu

Nimish A. Shah
Affiliation: Department of Mathematics, The Ohio State University, Columbus, Ohio 43210
Email: shah@math.ohio-state.edu

Keywords: Geometrically finite hyperbolic groups, mixing of geodesic flow, totally geodesic submanifolds, Patterson-Sullivan measure
Received by editor(s): April 7, 2011
Received by editor(s) in revised form: January 27, 2012, and May 31, 2012
Published electronically: October 2, 2012
Additional Notes: The first author was supported in part by NSF Grants #0629322 and #1068094.
The second author was supported in part by NSF Grant #1001654.
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.