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

Abstract

Given a topological orientable surface 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 struc- tures on S, which we call the Fenchel-Nielsen Teichmuller 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), (2) and (3), and we compared it to the classical Teichmuller met- ric (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.