## Nagoya Mathematical Journal

### A McShane-type identity for closed surfaces

Yi Huang

#### 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.

#### Article information

Source
Nagoya Math. J., Volume 219 (2015), 65-86.

Dates
Received: 11 October 2013
Accepted: 5 February 2014
First available in Project Euclid: 20 October 2015

Permanent link to this document
https://projecteuclid.org/euclid.nmj/1445345517

Digital Object Identifier
doi:10.1215/00277630-2887835

Mathematical Reviews number (MathSciNet)
MR3413573

Zentralblatt MATH identifier
1332.57017

Subjects
Primary: 57M50: Geometric structures on low-dimensional manifolds
Secondary: 57N05: Topology of $E^2$ , 2-manifolds

#### Citation

Huang, Yi. A McShane-type identity for closed surfaces. Nagoya Math. J. 219 (2015), 65--86. doi:10.1215/00277630-2887835. https://projecteuclid.org/euclid.nmj/1445345517

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