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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Equidistribution and counting for orbits of geometrically finite hyperbolic groups
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by Hee Oh and Nimish A. Shah PDF
J. Amer. Math. Soc. 26 (2013), 511-562 Request permission

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
  • 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.
  • © Copyright 2012 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • 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
  • MathSciNet review: 3011420