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Earthquakes in the length-spectrum Teichmüller spaces


Author: Dragomir Šarić
Journal: Proc. Amer. Math. Soc. 143 (2015), 1531-1543
MSC (2010): Primary 30F60; Secondary 32G15
DOI: https://doi.org/10.1090/S0002-9939-2014-12242-8
Published electronically: December 4, 2014
MathSciNet review: 3314067
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Abstract: Let $ X_0$ be a complete hyperbolic surface of infinite type that has a geodesic pants decomposition with cuff lengths bounded above. The length spectrum Teichmüller space $ T_{ls}(X_0)$ consists of homotopy classes of hyperbolic metrics on $ X_0$ such that the ratios of the corresponding simple closed geodesic for the hyperbolic metric on $ X_0$ and for the other hyperbolic metric are bounded from below away from 0 and from above away from $ \infty $. This paper studies earthquakes in the length spectrum Teichmüller space $ T_{ls}(X_0)$. We find a necessary condition and several sufficient conditions on the earthquake measure $ \mu $ such that the corresponding earthquake $ E^{\mu }$ describes a hyperbolic metric on $ X_0$ which is in the length spectrum Teichmüller space. Moreover, we give examples of earthquake paths $ t\mapsto E^{t\mu }$, for $ t\geq 0$, such that $ E^{t\mu }\in T_{ls}(X_0)$ for $ 0\leq t<t_0$, $ E^{t_0\mu }\notin T_{ls}(X_0)$ and $ E^{t\mu }\in T_{ls}(X_0)$ for $ t>t_0$.


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Dragomir Šarić
Affiliation: Department of Mathematics, Queens College of CUNY, 65-30 Kissena Boulevard, Flushing, New York 11367; and Mathematics Ph.D. Program, The CUNY Graduate Center, 365 Fifth Avenue, New York, New York 10016-4309
Email: Dragomir.Saric@qc.cuny.edu

DOI: https://doi.org/10.1090/S0002-9939-2014-12242-8
Received by editor(s): December 1, 2012
Received by editor(s) in revised form: April 24, 2013
Published electronically: December 4, 2014
Additional Notes: This research was partially supported by National Science Foundation grant DMS 1102440.
Communicated by: Michael Wolf
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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