Cell decompositions of Teichm\"uller spaces of surfaces with boundary

TL;DR

This paper extends the use of a family of coordinates for the Teichmüller space of surfaces with boundary, demonstrating that all such coordinates induce natural, mapping class group invariant cell decompositions.

Contribution

It generalizes previous results by showing that all coordinates h for h  0 produce invariant cell decompositions of the Teichmfcller space.

Findings

All h coordinates yield invariant cell decompositions.

The results apply to any h  0, extending prior work for h=0.

The decompositions are compatible with the mapping class group action.

Abstract

A family of coordinates $\psi_h$ for the Teichm\"uller space of a compact surface with boundary was introduced in \cite{l2}. In the work \cite{m1}, Mondello showed that the coordinate $\psi_0$ can be used to produce a natural cell decomposition of the Teichm\"uller space invariant under the action of the mapping class group. In this paper, we show that the similar result also works for all other coordinate $\psi_h$ for any $h \geq 0$.