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

Daniele Alessandrini, Lixin Liu, Athanase Papadopoulos, Weixu Su

Research output: Contribution to journalArticlepeer-review

4 Scopus citations

Abstract

Given a topological orientable surface S of finite or infinite type equipped with a pair of pants decomposition 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 Teichm̈uller space associated to the pair (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 [1-3], and we compared it with the classical Teichm̈uller 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.

Original languageEnglish
Pages (from-to)369-395
Number of pages27
JournalCommunications in Analysis and Geometry
Volume20
Issue number2
DOIs
StatePublished - Mar 2012

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