Nagoya Mathematical Journal

A McShane-type identity for closed surfaces

Yi Huang

Full-text: Open access


We prove a McShane-type identity: a series, expressed in terms of geodesic lengths, that sums to 2π for any closed hyperbolic surface with one distinguished point. To do so, we prove a generalized Birman–Series theorem showing that the set of complete geodesics on a hyperbolic surface with large cone angles is sparse.

Article information

Nagoya Math. J., Volume 219 (2015), 65-86.

Received: 11 October 2013
Accepted: 5 February 2014
First available in Project Euclid: 20 October 2015

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M50: Geometric structures on low-dimensional manifolds
Secondary: 57N05: Topology of $E^2$ , 2-manifolds

hyperbolic surfaces McShane identities geodesics


Huang, Yi. A McShane-type identity for closed surfaces. Nagoya Math. J. 219 (2015), 65--86. doi:10.1215/00277630-2887835.

Export citation


  • [1] J. S. Birman and C. Series, Geodesics with bounded intersection number on surfaces are sparsely distributed, Topology 24 (1985), 217–225.
  • [2] M. Bridgeman, Orthospectra of geodesic laminations and dilogarithm identities on moduli space, Geom. Topol. 15 (2011), 707–733.
  • [3] P. Buser, Geometry and Spectra of Compact Riemann Surfaces, Progr. Math. 106, Birkhäuser, Boston, 1992.
  • [4] F. Luo and S. P. Tan, A dilogarithm identity on moduli spaces of curves, J. Differential Geom. 97 (2014), 255–274.
  • [5] R. C. McOwen, Point singularities and conformal metrics on Riemann surfaces, Proc. Amer. Math. Soc. 103 (1988), 222–224.
  • [6] G. McShane. A remarkable identity for lengths of curves, Ph.D. dissertation, University of Warwick, Coventry, United Kingdom, 1991.
  • [7] G. McShane, Simple geodesics and a series constant over Teichmuller space, Invent. Math. 132 (1998), 607–632.
  • [8] G. McShane, Weierstrass points and simple geodesics, Bull. Lond. Math. Soc. 36 (2004), 181–187.
  • [9] G. McShane, Simple geodesics on surfaces of genus 2, Ann. Acad. Sci. Fenn. Math. 31 (2006), 31–38.
  • [10] M. Mirzakhani, Simple geodesics and Weil–Petersson volumes of moduli spaces of bordered Riemann surfaces, Invent. Math. 167 (2007), 179–222.
  • [11] M. Mirzakhani, Weil–Petersson volumes and intersection theory on the moduli space of curves, J. Amer. Math. Soc. 20 (2007), 1–23.
  • [12] G. Mondello, “Poisson structures on the Teichmüller space of hyperbolic surfaces with conical points” in In the Tradition of Ahlfors-Bers, V, Contemp. Math. 510, Amer. Math. Soc., Providence, 2010, 307–329.
  • [13] S. P. Tan, Length and trace series identities on Teichmüller spaces and character varieties, conference lecture at “Geometry, Topology and Dynamics of Character Varieties,” National University of Singapore, 2010.
  • [14] S. P. Tan, personal communication, August 2010.
  • [15] S. P. Tan, Y. L. Wong, and Y. Zhang, Generalizations of McShane’s identity to hyperbolic cone-surfaces, J. Differential Geom. 72 (2006), 73–112.
  • [16] M. Troyanov, Prescribing curvature on compact surfaces with conical singularities, Trans. Amer. Math. Soc. 324, no. 2 (1991), 793–821.
  • [17] E. Witten, “Two-dimensional gravity and intersection theory on moduli space” in Surveys in Differential Geometry (Cambridge, Mass., 1990), Lehigh University, Bethlehem, 1991, 243–310.