Geodesic
Equidecomposability, volume formulae and orthospectra
Masai, Hidetoshi1 ; McShane, Greg2
1Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, O-okayama, Meguro-ku, Tokyo 152-8552, Japan
2Institut Fourier, 100 rue des maths, BP 74, 38402 St Martin d’Hères, France
Algebraic and Geometric Topology, Tome 13 (2013) no. 6, pp. 3135-3152
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Bridgeman–Kahn and Calegari derived formulae for the volumes of compact hyperbolic n–manifolds with totally geodesic boundary in terms of the orthospectrum. Their methods are apparently different from each other, and involve computing the volume of different subspaces of unit tangent bundle of hyperbolic n–space. In this paper, we show that the two volume formulae coincide. We also derive a closed form of the formula in dimension 3.

DOI : 10.2140/agt.2013.13.3135
Classification : 57M50, 32Q45
Keywords: hyperbolic manifold, volume, orthospectrum

Masai, Hidetoshi  1   ; McShane, Greg  2

1 Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, O-okayama, Meguro-ku, Tokyo 152-8552, Japan
2 Institut Fourier, 100 rue des maths, BP 74, 38402 St Martin d’Hères, France 
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Masai, Hidetoshi; McShane, Greg. Equidecomposability, volume formulae and orthospectra. Algebraic and Geometric Topology, Tome 13 (2013) no. 6, pp. 3135-3152. doi: 10.2140/agt.2013.13.3135

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[4] D Calegari, Chimneys, leopard spots and the identities of Basmajian and Bridgeman, Algebr. Geom. Topol. 10 (2010) 1857

[5] D Calegari, Bridgeman's orthospectrum identity, Topology Proc. 38 (2011) 173

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[7] J G Ratcliffe, Foundations of hyperbolic manifolds, Graduate Texts in Mathematics 149, Springer (2006)

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