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

Athanase Papadopoulos (1, 2, 3) , Lixin Liu (3, 4) , Daniele Alessandrini (2) , Weixu Su (3, 4)

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

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.

Mots clés: Teichmüller space; Fenchel-Nielsen coordinates; Fenchel-Nielsen metric

Domaines: Topologie géométrique [math.GT]

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⟩