Generalizations of McShane's identity to hyperbolic cone-surfaces

Abstract

We generalize McShane's identity for the length series of simple closed geodesics on a cusped hyperbolic surface [19] to a general identity for hyperbolic cone-surfaces (with all cone angles ≤ π), possibly with cusps and/or geodesic boundary. The general identity is obtained by studying gaps formed by simple-normal geodesies emanating from a distinguished cone point, cusp or boundary geodesic. In particular, by applying the generalized identity to the quotient orbifolds of a hyperbolic one-cone/one-hole torus by its elliptic involution and of a hyperbolic closed genus two surface by its hyperelliptic involution, we obtain general Weierstrass identities for the one-cone/one-hole torus, and an identity for the genus two surface, which are also obtained by McShane using different methods in [20], [22] and [21]. We also give an interpretation of the general identity in terms of complex lengths of the cone points, cusps and geodesic boundary components.