Ergodicity of the geodesic flow on symmetric surfaces

TL;DR

This paper investigates conditions on Fenchel-Nielsen parameters that determine when a symmetric Riemann surface is of parabolic type, establishing new equivalences and criteria related to the surface's end symmetry and covering group properties.

Contribution

It provides new necessary and sufficient conditions for parabolicity of symmetric Riemann surfaces based on Fenchel-Nielsen coordinates, including an answer to an open question in the field.

Findings

Parabolicity characterized by Fenchel-Nielsen parameters for end symmetric surfaces.

Equivalence between parabolicity and the covering group being of the first kind for end symmetric surfaces.

Necessary and sufficient conditions for parabolicity in half-twist symmetric surfaces.

Abstract

We consider conditions on the Fenchel-Nielsen parameters of a Riemann surface $X$ that guarantee the surface $X$ is of parabolic type. An interesting class of Riemann surfaces for this problem is the one with finitely many topological ends. In this case the length part of the Fenchel-Nielsen coordinates can go to infinity for {parabolic $X$}. When the surface $X$ is end symmetric, we prove that {$X$ being parabolic} is equivalent to the covering group being of the first kind. Then we give necessary and sufficient conditions on the Fenchel-Nielsen coordinates of a half-twist symmetric surface $X$ such that {$X$ is parabolic}. As an application, we solve an open question from the prior work of Basmajian, Hakobyan and the second author.