Ergodicity of the geodesic flow on symmetric surfaces

Author:

Pandazis Michael,Šarić Dragomir

Abstract

We consider conditions on the Fenchel-Nielsen parameters of a Riemann surface X X that guarantee the surface X 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 X . When the surface X X is end symmetric, we prove that X 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 X such that X X is parabolic. As an application, we solve an open question from the prior work of Basmajian, Hakobyan and the second author [Proc. Lond. Math. Soc. (3) 125 (2022), pp. 568–625].

Funder

Simons Foundation

Publisher

American Mathematical Society (AMS)

Subject

Applied Mathematics,General Mathematics

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