## 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 $$(0,3)$$.

### MSC:

 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

### Keywords:

Fuchsian groups; McShane’s identity
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