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Geometric and Functional Analysis

, Volume 29, Issue 3, pp 871–889 | Cite as

Geodesic Currents and Counting Problems

  • Kasra RafiEmail author
  • Juan Souto
Article
  • 32 Downloads

Abstract

For every positive, continuous and homogeneous function f on the space of currents on a compact surface \({{{\overline{\Sigma }}}}\), and for every compactly supported filling current \(\alpha \), we compute as \(L \rightarrow \infty \), the number of mapping classes \(\phi \) so that \(f(\phi (\alpha ))\le L\). As an application, when the surface in question is closed, we prove a lattice counting theorem for Teichmüller space equipped with the Thurston metric.

Notes

Acknowledgements

The work of Maryam Mirzakhani provides the foundation for the results in this paper and in general she has been an inspiration for us. We dedicate this paper, which hopefully she would have found amusing, to her memory. We also thank the referee for helpful comments.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of TorontoTorontoCanada
  2. 2.IRMAR - Université de Rennes 1RennesFrance

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