Equidistribution of families of expanding horospheres on moduli spaces of hyperbolic surfaces

Abstract

Given a simple closed curve \(\gamma \) on a connected, oriented, closed surface S of negative Euler characteristic, Mirzakhani showed that the set of points in the moduli space of hyperbolic structures on S having a simple closed geodesic of length L of the same topological type as \(\gamma \) equidistributes with respect to a natural probability measure as \(L \rightarrow \infty \). We prove several generalizations of Mirzakhani’s result and discuss some of the technical aspects ommited in her original work. The dynamics of the earthquake flow play a fundamental role in the arguments in this paper.

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Acknowledgements

The author is very grateful to Alex Wright and Steven Kerckhoff for their invaluable advice, patience, and encouragement. The author would also like to thank Dat Pham Nguyen for very enlightening conversations.

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Correspondence to Francisco Arana-Herrera.

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Arana-Herrera, F. Equidistribution of families of expanding horospheres on moduli spaces of hyperbolic surfaces. Geom Dedicata 210, 65–102 (2021). https://doi.org/10.1007/s10711-020-00534-6

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Keywords

  • Equidistribution
  • Expanding
  • Horospheres
  • Moduli spaces
  • Hyperbolic surfaces
  • Teichmüller theory
  • Earthquake flow
  • Random Riemann surfaces

Mathematics Subject Classification

  • 30F60
  • 32G15