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Chimneys, leopard spots and the identities of Basmajian and Bridgeman

Calegari, Danny (2010) Chimneys, leopard spots and the identities of Basmajian and Bridgeman. Algebraic and Geometric Topology, 10 (3). pp. 1857-1863. ISSN 1472-2747.

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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) = ∑_(i) q_(n)(e^l_i) where the sum is taken over the orthospectrum of M. When n = 3, this has the explicit form ∑_(i) 1/(e^(2l_(i)) − 1) = −χ(S)/4.

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Additional Information:© 2010 Mathematical Sciences Publishers. Received: 26 May 2010, Revised: 26 July 2010, Accepted: 28 July 2010, Published: 3 September 2010. I would like to thank Martin Bridgeman, Jeremy Kahn, Sadayoshi Kojima, Greg McShane and Maryam Mirzakhani for some useful discussions. In particular, this paper owes an obvious debt to [2] and the beautiful sequel [3]. Thanks also to the referee for a useful correction. The first version of this paper was written before the author was aware of [1], and I am very grateful to Greg and Sadayoshi for bringing it to my attention. The author was supported by NSF grant DMS 0707130.
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NSFDMS 0707130
Issue or Number:3
Classification Code:Primary 57M50; Secondary 11J06
Record Number:CaltechAUTHORS:20101122-112850841
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:20947
Deposited By: Benjamin Perez
Deposited On:22 Nov 2010 23:43
Last Modified:03 Oct 2019 02:17

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