A McShane-type identity for closed surfaces

Yi Huang

doi:10.1215/00277630-2887835

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Home > Journals > Nagoya Math. J. > Volume 219 > Article

September 2015 A McShane-type identity for closed surfaces

Yi Huang

Nagoya Math. J. 219: 65-86 (September 2015). DOI: 10.1215/00277630-2887835

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Abstract

We prove a McShane-type identity: a series, expressed in terms of geodesic lengths, that sums to $2\pi$ 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.