Geometry & Topology

Orthospectra of geodesic laminations and dilogarithm identities on moduli space

Martin Bridgeman

Full-text: Open access


Given a measured lamination λ on a finite area hyperbolic surface we consider a natural measure Mλ on the real line obtained by taking the push-forward of the volume measure of the unit tangent bundle of the surface under an intersection function associated with the lamination. We show that the measure Mλ gives summation identities for the Rogers dilogarithm function on the moduli space of a surface.

Article information

Geom. Topol., Volume 15, Number 2 (2011), 707-733.

Received: 27 December 2010
Accepted: 14 February 2011
First available in Project Euclid: 20 December 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 32G15: Moduli of Riemann surfaces, Teichmüller theory [See also 14H15, 30Fxx]
Secondary: 11M36: Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. Explicit formulas



Bridgeman, Martin. Orthospectra of geodesic laminations and dilogarithm identities on moduli space. Geom. Topol. 15 (2011), no. 2, 707--733. doi:10.2140/gt.2011.15.707.

Export citation


  • A Basmajian, The orthogonal spectrum of a hyperbolic manifold, Amer. J. Math. 115 (1993) 1139–1159
  • B C Berndt, Ramanujan's Notebooks Part IV, Springer, New York (1994)
  • M Bridgeman, D Dumas, Distribution of intersection lengths of a random geodesic with a geodesic lamination, Ergodic Theory Dynam. Systems 27 (2007) 1055–1072
  • M Bridgeman, J Kahn, Hyperbolic volume of manifolds with geodesic boundary and orthospectra, Geom. Funct. Anal. 20 (2010) 1210–1230
  • D Calegari, Chimneys, leopard spots and the identities of Basmajian and Bridgeman, Algebr. Geom. Topol. 10 (2010) 1857–1863
  • D Calegari, Bridgeman's orthospectrum identity, Topology Proc. 38 (2011) 173–179
  • B Gordon, R J McIntosh, Algebraic dilogarithm identities, Ramanujan J. 1 (1997) 431–448 International Symposium on Number Theory (Madras, 1996)
  • E Hopf, Statistik der geodätischen Linien in Mannigfaltigkeiten negativer Krümmung, Ber. Verh. Sächs. Akad. Wiss. Leipzig 91 (1939) 261–304
  • L Lewin (editor), Structural properties of polylogarithms, Mathematical Surveys and Monographs 37, American Mathematical Society, Providence, RI (1991)
  • P J Nicholls, The ergodic theory of discrete groups, London Mathematical Society Lecture Note Series 143, Cambridge University Press, Cambridge (1989)
  • L J Rogers, On function sum theorems connected with the series $\sum_1^{\infty} \frac{x^n}{n^2}$, Proc. London Math. Soc. 4 (1907) 169–189