Earthquakes in the length-spectrum Teichmüller spaces

Šarić, Dragomir

doi:10.1090/S0002-9939-2014-12242-8

Earthquakes in the length-spectrum Teichmüller spaces
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Proc. Amer. Math. Soc. 143 (2015), 1531-1543 Request permission

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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