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 $2\left(n-2\right)$ 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 ${\sum }_{i}1∕\left(e^{2l_i}-1\right)=-\chi \left(S\right)∕4$.