Chimneys, leopard spots, and the identities of Basmajian and Bridgeman

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 q_n of degree 2(n-2) so that if M is a compact hyperbolic manifold of dimension n with totally geodesic boundary S, there is an identity χ(S) = \sum_i q_n(e^{l_i}) where the sum is taken over the orthospectrum of M. When n=3, this has the explicit form \sum_i 1/(e^{2l_i}-1) = -χ(S)/4.

6 pages; version 2 incorporates referee's comments

References in corpus (2)

• Bridgeman's orthospectrum identity
• Hyperbolic volume of n-manifolds with geodesic boundary and orthospectra

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• Geodesic and orthogeodesic identities on hyperbolic surfaces
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• Lower bounds for volumes and orthospectra of hyperbolic manifolds with geodesic boundary
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• The Bridgeman-Kahn identity for hyperbolic manifolds with cusped boundary
• On systoles and ortho spectrum rigidity
• The Brownian loop measure on Riemann surfaces and applications to length spectra
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