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 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 geodesics 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. We also give an interpretation of the general identity in terms of complex lengths of the cone points, cusps and geodesic boundary components.