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Chimneys, leopard spots and the identities of Basmajian and Bridgeman

Danny Calegari

Algebraic & Geometric Topology 10 (2010) 1857–1863
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(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) = iqn(eli) where the sum is taken over the orthospectrum of M. When n = 3, this has the explicit form i1(e2li 1) = χ(S)4.

Keywords
orthospectrum, identity, chimney, leopard spot, dilogarithm
Mathematical Subject Classification 2000
Primary: 57M50
Secondary: 11J06
References
Publication
Received: 26 May 2010
Revised: 26 July 2010
Accepted: 28 July 2010
Published: 3 September 2010
Authors
Danny Calegari
Department of Mathematics
Caltech
Pasadena CA, 91125
http://www.its.caltech.edu/~dannyc