Chimneys, leopard spots, and the identities of Basmajian and Bridgeman

TL;DR

This paper presents a simple geometric approach to derive and simplify orthospectrum identities for hyperbolic manifolds, providing explicit formulas and rational functions for various dimensions.

Contribution

It introduces a unified geometric method to derive orthospectrum identities and explicitly determines summands using rational functions for different dimensions.

Findings

Derived orthospectrum identities for hyperbolic manifolds.

Explicit rational functions for summands in identities.

Simplified the determination of summands in these identities.

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 \chi(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^{2l_i}-1) = -\chi(S)/4.