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

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1 IRMA - Institut de Recherche Mathématique Avancée 2 MPI - Max-Plank-Institut für Mathematik 3 Hausdorff Center for Mathematics and Institute for Numerical Simulation - University of Bonn 4 Department of Mathematics

Abstract : Given a topological orientable surface 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}, \cite{various} and \cite{local}, and we compared it to the classical Teichmüller metric defined using quasi-conformal mappings and to another metric, namely, the length spectrum, defined using ratios of hyperbolic lengths of simple closed curves metric. 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. The results complement results obtained in the previous papers and they show that these previous results are optimal.

keyword : Fenchel-Nielsen metric Teichmüller space Fenchel-Nielsen coordinates

Autor: Athanase Papadopoulos - Lixin Liu - Daniele Alessandrini - Weixu Su -

Fuente: https://hal.archives-ouvertes.fr/

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