Generalizations of McShane's identity to hyperbolic cone-surfaces

By S. Tan, Yan Loi Wong, Ying Zhang. 2004

79 citations recorded by the source corpus.

We generalize McShane’s identity for the length series of simple closed geodesics on a cusped hyperbolic surface [19] 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 simplenormal 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 in [20], [22] and [21]. We also give an interpretation of the general identity in terms of complex lengths of the cone points, cusps and geodesic boundary components.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/faeddab95ca5fae2c5b8fb5655901210e9ac94d3

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In Geometry and Topology

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Citations in the corpus

Low-dimensional de Sitter quantum gravity (2019, background): “p at which a point is removed. This cusped surface is the same one that arises in the !0 limit.8 A stronger piece of evidence, which in fact subsumes this observation, comes from Tan, Wong, and Zhang [52]. They proved that the conjecture Eq. (3.28) holds as long as i1=2, i.e. the cone angles are les”
Families of Riemann Surfaces and Weil-Petersson Geometry (2010, background): “[TWZ06a] Ser Peow Tan, Yan Loi Wong, and Ying Zhang.”
The path integral of 3D gravity near extremality; or, JT gravity with defects as a matrix integral (2020, background): “Fortunately, the relevant volumes have been studied already [33, 34, 35], and they are obtained simply from the usual WP volumes by inserting a boundary with b = 2πiα for each defect (proven for α ≤ 1 2 , but conjectured more generally).”
Generalized Markoff maps and McShane's identity (2005, methodology): “The only example in the surface case which we know of is an identity for the genus two surface (see [20] and [29]).”
Geodesics with one self-intersection, and other stories (2009, background): “ifthesurfaceisstillatorus,butinsteadofacuspitnowpossesses a boundary component of length δor cone angle θ.This case was also considered by G. McShane, but, to my knowledge, ﬁrst appeared in the paper [42]. We will summarize all the results in the two theorems below: Theorem 3.5 (McShane’s ”classic” ”