Communications in Analysis and Geometry

Volume 20 (2012)

Number 2

The behaviour of Fenchel–Nielsen distance under a change of pants decomposition

Pages: 369 – 395



Daniele Alessandrini (Max-Planck-Institut für Mathematik, Bonn, Germany)

Lixin Liu (Department of Mathematics, Sun-Yat-Sen (Zhongshan) University, Guangzhou, China)

Athanase Papadopoulos (CNRS, Université de Strasbourg, France)

Weixu Su (Department of Mathematics, Fudan University, Shanghai, China)


Given a topological orientable surface $S$ of finite or infinite type equipped with a pair of pants decomposition $\mathcal{P}$ and given a base complex structure $X$ on $S$, there is an associated deformation space of complex structures on $S$, which we call the Fenchel–Nielsen Teichmüller space associated to the pair $(\mathcal{P},X)$. This space carries a metric, which we call the Fenchel–Nielsen metric, defined using Fenchel–Nielsen coordinates. We studied this metric in the papers \cite{ALPSS,various,local}, and we compared it with the classical Teichmüller metric (defined using quasi-conformal mappings) and to the length spectrum metric (defined using ratios of hyperbolic lengths of simple closed curves). In the present paper, we show that under a change of pair of pants decomposition, the identity map between the corresponding Fenchel–Nielsen metrics is not necessarily bi-Lipschitz. These results complement results obtained in the previous papers and they show that these previous results are optimal.

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