The type problem for Riemann surfaces via Fenchel–Nielsen parameters

Abstract

A Riemann surface X $X$ is said to be of parabolic type if it does not support a Green's function. Equivalently, the geodesic flow on the unit tangent bundle of X $X$ (equipped with the hyperbolic metric) is ergodic. Given a Riemann surface X $X$ of arbitrary topological type and a hyperbolic pants decomposition of X $X$ , we obtain sufficient conditions for parabolicity of X $X$ in terms of the Fenchel–Nielsen parameters of the decomposition. In particular, we initiate the study of the effect of twist parameters on parabolicity. A key ingredient in our work is the notion of nonstandard half-collar about a hyperbolic geodesic. We show that the modulus of such a half-collar is much larger than the modulus of a standard half-collar as the hyperbolic length of the core geodesic tends to infinity. Moreover, the modulus of the annulus obtained by gluing two nonstandard half-collars depends on the twist parameter, unlike in the case of standard collars. Our results are sharp in many cases. For instance, for zero-twist flute surfaces as well as for half-twist flute surfaces with concave sequences of lengths our results provide a complete characterization of parabolicity in terms of the length parameters. It follows that parabolicity is equivalent to completeness in these cases. Applications to other topological types such as surfaces with infinite genus and one end (also known as the infinite Loch–Ness monster), the ladder surface, and abelian covers of compact surfaces are also studied.