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Inequivalent topologies on the Teichmüller space of the flute surface


Author: Özgür Evren
Journal: Proc. Amer. Math. Soc. 145 (2017), 2607-2621
MSC (2010): Primary 30F60
DOI: https://doi.org/10.1090/proc/13412
Published electronically: December 27, 2016
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Abstract: The topology defined by the symmetrized Lipschitz metric on the Teichmüller space of an infinite type surface, in contrast to finite type surfaces, need not be the same as the topology defined by the Teichmüller metric. In this paper, we study the equivalence of these topologies on a particular kind of infinite type surface, called the flute surface.

Following a construction by Shiga and using additional hyperbolic geometric estimates, we obtain sufficient conditions in terms of length parameters for these two metrics to be topologically inequivalent. Next, we construct infinite parameter families of quasiconformally distinct flute surfaces with the property that the symmetrized Lipschitz metric is not topologically equivalent to the Teichmüller metric.


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Additional Information

Özgür Evren
Affiliation: Department of Mathematics, Galatasaray University, Istanbul, Turkey
Email: oevren@gradcenter.cuny.edu

DOI: https://doi.org/10.1090/proc/13412
Keywords: The symmetrized Lipschitz metric, infinite type Riemann surfaces
Received by editor(s): September 25, 2015
Received by editor(s) in revised form: July 29, 2016
Published electronically: December 27, 2016
Additional Notes: The author was supported by Tübitak through 1001 - Scientific and Technological Research Projects Funding Program, project 114R073.
Communicated by: Ken Ono
Article copyright: © Copyright 2016 American Mathematical Society