Mathematische Zeitschrift

, Volume 279, Issue 3–4, pp 1175–1196 | Cite as

On the hyperbolic orbital counting problem in conjugacy classes

Article

Abstract

Given a discrete group \(\Gamma \) of isometries of a negatively curved manifold \({\widetilde{M}}\), a non-trivial conjugacy class \({\mathfrak {K}}\) in \(\Gamma \) and \(x_0\in {\widetilde{M}}\), we give asymptotic counting results, as \(t\rightarrow +\infty \), on the number of orbit points \(\gamma x_0\) at distance at most \(t\) from \(x_0\), when \(\gamma \) is restricted to be in \({\mathfrak {K}}\), as well as related equidistribution results. These results generalise and extend work of Huber on cocompact hyperbolic lattices in dimension 2. We also study the growth of given conjugacy classes in finitely generated groups endowed with a word metric.

Keywords

Counting Equidistribution Hyperbolic geometry Growth Conjugacy class 

Mathematics Subject Classification

37C35 20H10 30F40 53A35 20G20 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of JyväskyläUniversity of JyväskyläFinland
  2. 2.Département de mathématique, UMR 8628 CNRS, Bât. 425Université Paris-SudOrsay CedexFrance

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