Earthquakes in the length-spectrum Teichm\"uller spaces

TL;DR

This paper investigates earthquakes in the length-spectrum Teichmüller space of infinite-type hyperbolic surfaces, providing conditions for when earthquake deformations stay within this space and analyzing their boundary behavior.

Contribution

It establishes necessary and sufficient conditions for earthquakes to remain in the length spectrum Teichmüller space and constructs examples illustrating boundary phenomena.

Findings

Identifies conditions on earthquake measures for space inclusion

Constructs earthquake paths crossing the boundary of the space

Provides examples of boundary behavior of earthquake deformations

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 Teichm\"uller 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 the below away from 0 and from the above away from $\infty$ (cf. \cite{ALPS}). This paper studies earthquakes in the length spectrum Teichm\"uller space $T_{ls}(X_0)$. We find a necessary condition and several sufficient conditions on earthquake measure $\mu$ such that the corresponding earthquake $E^{\mu}$ describes the hyperbolic metric on $X_0$ which is in the length spectrum Teichm\"uller 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 $t>t_0$.