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
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 $2\left(n-2\right)$ so that if $M$ is a compact hyperbolic manifold of dimension $n$ with totally geodesic boundary $S$, there is an identity $\chi \left(S\right)={\sum }_{i}{q}_{n}\left({e}^{{l}_{i}}\right)$ where the sum is taken over the orthospectrum of $M$. When $n=3$, this has the explicit form ${\sum }_{i}1∕\left({e}^{2{l}_{i}}-1\right)=-\chi \left(S\right)∕4$.

Keywords
orthospectrum, identity, chimney, leopard spot, dilogarithm
Primary: 57M50
Secondary: 11J06
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