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

TL;DR

This paper investigates how the Fenchel-Nielsen metric on Teichmüller space behaves under changes in pants decompositions, revealing that the identity map between different Fenchel-Nielsen metrics may not be bi-Lipschitz, thus highlighting limitations of previous assumptions.

Contribution

It demonstrates that changing pants decompositions can lead to non-bi-Lipschitz equivalences between Fenchel-Nielsen metrics, showing the optimality of earlier results.

Findings

Identity map between Fenchel-Nielsen metrics may not be bi-Lipschitz after changing decompositions.

Previous results on metric comparisons are shown to be optimal.

The study clarifies the limitations of Fenchel-Nielsen metric invariance under decomposition changes.

Abstract

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 Teichm\"uller 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 Teichm\"uller 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.