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Proceedings of the American Mathematical Society

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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
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

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.