The type problem for Riemann surfaces via Fenchel–Nielsen parameters

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 unit tangent bundle (equipped with hyperbolic metric) ergodic. Given arbitrary topological and pants decomposition , we obtain sufficient conditions for parabolicity in terms Fenchel–Nielsen parameters decomposition. In particular, initiate study effect twist parabolicity. key ingredient our work notion nonstandard half-collar about geodesic. We show that modulus such much larger than standard as length core tends infinity. Moreover, annulus obtained by gluing two half-collars depends parameter, unlike case collars. Our results are sharp many cases. For instance, zero-twist flute surfaces well half-twist concave sequences lengths provide complete characterization parameters. It follows equivalent completeness these Applications other types infinite genus one end (also known Loch–Ness monster), ladder surface, abelian covers compact also studied.