Fenchel-Nielsen coordinates for asymptotically conformal deformations

Abstract

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.