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

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

2010-05-26

arxiv.org/abs/1005.4858v2

Algebr. Geom. Topol. 10 (2010), no. 3, 1857-1863

Mathematics

Geometric Topology

6 pages; version 2 incorporates referee's comments

Scientific paper

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.

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