Ergodicity of the geodesic flow on symmetric surfaces
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- by Michael Pandazis and Dragomir Šarić;
- 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].References
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Bibliographic Information
- Michael Pandazis
- Affiliation: PhD Program in Mathematics, The Graduate Center, CUNY, 365 Fifth Ave., New York, New York 10016
- ORCID: 0000-0001-9313-2755
- Email: mpandazis@gradcenter.cuny.edu
- Dragomir Šarić
- Affiliation: PhD Program in Mathematics, The Graduate Center, CUNY, 365 Fifth Ave., New York, New York 10016; and Department of Mathematics, Queens College, CUNY, 65–30 Kissena Blvd., Flushing, New York 11367
- MR Author ID: 695566
- Email: Dragomir.Saric@qc.cuny.edu
- Received by editor(s): November 30, 2022
- Received by editor(s) in revised form: February 10, 2023
- Published electronically: May 19, 2023
- Additional Notes: The second author was partially supported by the Simons Foundation Collaboration Grant 346391 and by PSCCUNY grants.
- © Copyright 2023 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 376 (2023), 7013-7043
- MSC (2020): Primary 30F20, 30F25, 30F45, 57K20
- DOI: https://doi.org/10.1090/tran/8924
- MathSciNet review: 4636683