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

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.