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Ergodicity of the geodesic flow on symmetric surfaces
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Trans. Amer. Math. Soc. 376 (2023), 7013-7043

DOI: https://doi.org/10.1090/tran/8924

Published electronically: May 19, 2023

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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 [Proc. Lond. Math. Soc. (3) 125 (2022), pp. 568–625].

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