An infinite product associated to a hyperbolic three-holed sphere. (English) Zbl 1300.30080

Summary: One of the generalizations of McShane’s identities by Tan, Wong and Zhang is an identity concerning lengths of simple closed geodesics which pass through two Weierstrass points on a hyperbolic one-holed torus. The Fuchsian groups which uniformize the surface are purely hyperbolic and free of rank two. Another type of Fuchsian groups of the same property is of type \((0,3)\) corresponding to hyperbolic three-holed spheres. In this paper we show a McShane-type identity which holds for all Fuchsian groups of type \((0,3)\).


30F35 Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization)
20H10 Fuchsian groups and their generalizations (group-theoretic aspects)
57M50 General geometric structures on low-dimensional manifolds
Full Text: Euclid