GENERALIZATIONS OF MCSHANE ’ S IDENTITY TO HYPERBOLIC CONE - SURFACES 3 Theorem 1

We generalize McShane’s identity for the length series of simple closed geodesics on a cusped hyperbolic surface [17] to hyperbolic cone-surfaces (with all cone angles ≤ π), 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 in [18], [20] and [19]. We also give an interpretation of the identity in terms of complex lengths, gaps, and the direct visual measure of the boundary.