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(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.