Algebraic & Geometric Topology

Chimneys, leopard spots and the identities of Basmajian and Bridgeman

Danny Calegari

Full-text: Open access

Abstract

We give a simple geometric argument to derive in a common manner orthospectrum identities of Basmajian and Bridgeman. Our method also considerably simplifies the determination of the summands in these identities. For example, for every odd integer n, there is a rational function qn of degree 2(n2) so that if M is a compact hyperbolic manifold of dimension n with totally geodesic boundary S, there is an identity χ(S)=iqn(eli) where the sum is taken over the orthospectrum of M. When n=3, this has the explicit form i1(e2li1)=χ(S)4.

Article information

Source
Algebr. Geom. Topol., Volume 10, Number 3 (2010), 1857-1863.

Dates
Received: 26 May 2010
Revised: 26 July 2010
Accepted: 28 July 2010
First available in Project Euclid: 19 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513715155

Digital Object Identifier
doi:10.2140/agt.2010.10.1857

Mathematical Reviews number (MathSciNet)
MR2684984

Zentralblatt MATH identifier
1196.57010

Subjects
Primary: 57M50: Geometric structures on low-dimensional manifolds
Secondary: 11J06: Markov and Lagrange spectra and generalizations

Keywords
orthospectrum identity chimney leopard spot dilogarithm

Citation

Calegari, Danny. Chimneys, leopard spots and the identities of Basmajian and Bridgeman. Algebr. Geom. Topol. 10 (2010), no. 3, 1857--1863. doi:10.2140/agt.2010.10.1857. https://projecteuclid.org/euclid.agt/1513715155


Export citation

References

  • A Basmajian, The orthogonal spectrum of a hyperbolic manifold, Amer. J. Math. 115 (1993) 1139–1159
  • M Bridgeman, Orthospectra of geodesic laminations and dilogarithm identities on moduli space
  • M Bridgeman, J Kahn, Hyperbolic volume of $n$–manifolds with geodesic boundary and orthospectra, to appear in Geom. Funct. Anal.
  • D Calegari, Bridgeman's orthospectrum identity, to appear in Topol. Proc.
  • G McShane, Simple geodesics and a series constant over Teichmuller space, Invent. Math. 132 (1998) 607–632
  • M Mirzakhani, Simple geodesics and Weil–Petersson volumes of moduli spaces of bordered Riemann surfaces, Invent. Math. 167 (2007) 179–222
  • J-P Otal, Thurston's hyperbolization of Haken manifolds, from: “Surveys in differential geometry, Vol. III (Cambridge, MA, 1996)”, (C-C Hsiung, S-T Yau, editors), Int. Press, Boston (1998) 77–194