Earthquakes in the length-spectrum Teichm\"uller spaces

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 $t>t_0$.Comment: metadata correction, the same version as befor

Bendings by finitely additive transverse cocycles

Let $S$ be any closed hyperbolic surface and let $\lambda$ be a maximal geodesic lamination on $S$. The amount of bending of an abstract pleated surface (homeomorphic to $S$) with the pleating locus $\lambda$ is completely determined by an $(\mathbb{R}/2\pi\mathbb{Z})$-valued finitely additive transverse cocycle $\beta$ to the geodesic lamination $\lambda$. We give a sufficient condition on $\beta$ such that the corresponding pleating map $\tilde{f}_{\beta}:\mathbb{H}^2\to\mathbb{H}^3$ induces a quasiFuchsian representation of the surface group $\pi_1(S)$. Our condition is genus independent.Comment: 34 pages, 4 figures, extra explanations added, same theorem

A Thurston boundary for infinite-dimensional Teichm\"uller spaces

For a compact surface $X_0$, Thurston introduced a compactification of its Teichm\"uller space $\mathcal T(X_0)$ by completing it with a boundary $\mathcal{PML}(X_0)$ consisting of projective measured geodesic laminations. We introduce a similar bordification for the Teichm\"uller space $\mathcal T(X_0)$ of a noncompact Riemann surface $X_0$, using the technical tool of geodesic currents. The lack of compactness requires the introduction of certain uniformity conditions which were unnecessary for compact surfaces. A technical step, providing a convergence result for earthquake paths in $\mathcal T(X_0)$, may be of independent interest.Comment: 42 pages, 3 figure

Quadratic differentials and foliations on infinite Riemann surfaces

We prove that an infinite Riemann surface $X$ is parabolic ($X\in O_G$) if and only if the union of the horizontal trajectories of any integrable holomorphic quadratic differential that are cross-cuts is of zero measure. Then we establish the density of the Jenkins-Strebel differentials in the space of all integrable quadratic differentials when $X\in O_G$ and extend Kerckhoff's formula for the Teichm\"uller metric in this case. Our methods depend on extending to infinite surfaces the Hubbard-Masur theorem describing which measured foliations can be realized by horizontal trajectories of integrable holomorphic quadratic differentials.Comment: 41 pages, 9 figure

The Teichmüller distance between finite index subgroups of PSL_2ℤ

For a given ϵ>0, we show that there exist two finite index subgroups of PSL_2(ℤ) which are (1+ϵ)-quasisymmetrically conjugated and the conjugation homeomorphism is not conformal. This implies that for any ϵ>0 there are two finite regular covers of the Modular once punctured torus T_0 (or just the Modular torus) and a (1+ϵ)-quasiconformal map between them that is not homotopic to a conformal map. As an application of the above results, we show that the orbit of the basepoint in the Teichmüller space T(S^p) of the punctured solenoid S^p under the action of the corresponding Modular group (which is the mapping class group of S^p [6], [7]) has the closure in T(S^p) strictly larger than the orbit and that the closure is necessarily uncountable