Generalizations of McShane's identity to hyperbolic cone-surfaces

Details Generalizations of McShane's identity to hyperbolic cone-surfaces Generalizations of McShane's identity to hyperbolic cone-surfaces

2004-04-12

arxiv.org/abs/math/0404226v1

J. Differential Geom., 72 (2006) 73-112

Mathematics

Geometric Topology

37 pages, 9 figures

Scientific paper

We generalize McShane's identity for the length series of simple closed geodesics on a cusped hyperbolic surface to hyperbolic cone-surfaces (with all cone angles $\le \pi$), possibly with cusps and/or geodesic boundary. In particular, by applying the generalized identity to the orbifolds obtained from taking the quotient of the one-holed torus by its elliptic involution, and the closed genus two surface by its hyper-elliptic involution, we obtain generalizations of the Weierstrass identities for the one-holed torus, and identities for the genus two surface, also obtained by McShane using different methods. We also give an interpretation of the identity in terms of complex lengths, gaps, and the direct visual measure of the boundary.

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