Algebraic & Geometric Topology Volume 10, issue 3 (2010)

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Chimneys, leopard spots and the identities of Basmajian and Bridgeman

Danny Calegari

Algebraic & Geometric Topology 10 (2010) 1857–1863

DOI: 10.2140/agt.2010.10.1857

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 $\sum_{i} 1 ∕ (e^{2 l_{i}} - 1) = - χ (S) ∕ 4$ .

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