Fenchel-Nielsen coordinates for asymptotically conformal deformations

Let $X$ be an infinite hyperbolic surface endowed with an upper bounded geodesic pants decomposition. Alessandrini, Liu, Papadopoulos, Su and Sun \cite{ALPSS}, \cite{ALPS} parametrized the quasiconformal Teichm\"uller space $T_{qc}(X)$ and the length spectrum Teichm\"uller space $T_{ls}(X)$ using the Fenchel-Nielsen coordinates. A quasiconformal map $f:X\to Y$ is said to be {\it asymptotically conformal} if its Beltrami coefficient $\mu =\bar{\partial}f/\partial f$ converges to zero at infinity. The space of all asymptotically conformal maps up to homotopy and post-composition by conformal maps is called "little" Teichm\"uller space $T_0(X)$. We find a parametrization of $T_0(X)$ using the Fenchel-Nielsen coordinates and a parametrization of the closure $\overline{T_0(X)}$ of $T_0(X)$ in the length spectrum metric. We also prove that the quotients $AT(X)=T_{qc}(X)/T_0(X)$, $T_{ls}(X)/\overline{T_{qc}(X)}$ and $T_{ls}(X)/\overline{T_0(X)}$ are contractible in the Teichm\"uller metric and the length spectrum metric, respectively. Finally, we show that the Wolpert's lemma on the lengths of simple closed geodesics under quasiconformal maps is not sharp.Comment: 10 pages, 1 figur

Infinitesimal Liouville currents, cross-ratios and intersection numbers

Many classical objects on a surface S can be interpreted as cross-ratio functions on the circle at infinity of the universal covering. This includes closed curves considered up to homotopy, metrics of negative curvature considered up to isotopy and, in the case of interest here, tangent vectors to the Teichm\"uller space of complex structures on S. When two cross-ratio functions are sufficiently regular, they have a geometric intersection number, which generalizes the intersection number of two closed curves. In the case of the cross-ratio functions associated to tangent vectors to the Teichm\"uller space, we show that two such cross-ratio functions have a well-defined geometric intersection number, and that this intersection number is equal to the Weil-Petersson scalar product of the corresponding vectors.Comment: 17 page