Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 41, 2016, 167-176

FENCHEL-NIELSEN COORDINATES FOR ASYMPTOTICALLY CONFORMAL DEFORMATIONS

Dragomir Šaric

Queens College of CUNY, Department of Mathematics
65-30 Kissena Blvd., Flushing, NY 11367, U.S.A.; Dragomir.Saric 'at' qc.cuny.edu
and The CUNY Graduate Center, Mathematics PhD Program
365 Fifth Avenue, New York, NY 10016-4309, U.S.A.

Abstract. Let X be an infinite hyperbolic surface endowed with an upper bounded geodesic pants decomposition. Alessandrini, Liu, Papadopoulos, Su and Sun [2,3] parametrized the quasiconformal Teichmüller space T_qc(X) and the length spectrum Teichmüller space T_ls(X) using the Fenchel-Nielsen coordinates. A quasiconformal map f : X → Y is said to be asymptotically conformal if its Beltrami coefficient μ = ^-∂f/∂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" Teichmüller space T₀(X). We find a parametrization of T₀(X) using the Fenchel-Nielsen coordinates and a parametrization of the closure \overline{T₀(X)} of T₀(X) in the length spectrum metric. We also prove that the quotients AT(X) = T_qc(X)/T₀(X), T_ls(X)/\overline{T_qc(X)} and T_ls(X)/\overline{T₀(X)} are contractible in the Teichmüller 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.

2010 Mathematics Subject Classification: Primary 30F60, 32G15.

Key words: Teichmüller, asymptotically conformal, infinite surfaces, geodesic lenghts.

Reference to this article: D. Šaric: Fenchel-Nielsen coordinates for asymptotically conformal deformations. Ann. Acad. Sci. Fenn. Math. 41 (2016), 167-176.

Full document as PDF file

doi:10.5186/aasfm.2016.4112