A McShane-type identity for closed surfaces

TL;DR

This paper establishes a McShane-type identity for closed hyperbolic surfaces with a distinguished point, linking geodesic lengths to a constant sum, and introduces a generalized Birman-Series theorem for surfaces with large cone angles.

Contribution

It presents a novel McShane-type identity for closed surfaces and proves a generalized Birman-Series theorem for hyperbolic surfaces with large cone angles.

Findings

Series sums to 2π for closed hyperbolic surfaces with a distinguished point

Set of complete geodesics with large cone angles is sparse

Generalized Birman-Series theorem proved

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.