We prove a McShane-type identity: a series, expressed in terms of geodesic lengths, that sums to for any closed hyperbolic surface with one distinguished point. To do so, we prove a generalized Birman–Series theorem showing that the set of complete geodesics on a hyperbolic surface with large cone angles is sparse.
"A McShane-type identity for closed surfaces." Nagoya Math. J. 219 65 - 86, September 2015. https://doi.org/10.1215/00277630-2887835