Article Dans Une Revue Communications in Analysis and Geometry Année : 2012

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

Résumé

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.

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hal-00589706 , version 1 (30-04-2011)

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Athanase Papadopoulos, Lixin Liu, Daniele Alessandrini, Weixu Su. The behaviour of Fenchel-Nielsen distance under a change of pants decomposition. Communications in Analysis and Geometry, 2012, 20 (2), p. 369-395. ⟨10.4310/CAG.2012.v20.n2.a6⟩. ⟨hal-00589706⟩
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