Hyperbolic metrics, measured foliations and pants decompositions for non-orientable surfaces

TL;DR

This paper extends fundamental theorems of Thurston theory from orientable to non-orientable surfaces, introducing new parametrizations and moves for pants decompositions and measured foliations.

Contribution

It provides non-orientable analogues of key theorems in Thurston theory, including parametrizations of Teichmüller space and measured foliations, with new elementary moves introduced.

Findings

Analogues of Fenchel-Nielsen and Dehn-Thurston theorems for non-orientable surfaces.

New elementary moves on pants decompositions for non-orientable surfaces.

Drop of twisting number for 1-sided curves in pants decompositions.

Abstract

We provide analogues for non-orientable surfaces with or without boundary or punctures of several basic theorems in the setting of the Thurston theory of surfaces which were developed so far only in the case of orientable surfaces. Namely, we provide natural analogues for non-orientable surfaces of the Fenchel-Nielsen theorem on the parametrization of the Teichm\"uller space of the surface, the Dehn-Thurston theorem on the parametrization of measured foliations in the surface, and the Hatcher-Thurston theorem, which gives a complete minimal set of moves between pair of pants decompositions of the surface. For the former two theorems, one in effect drops the twisting number for any curve in a pants decomposition which is 1-sided, and for the latter, new elementary moves on pants decompositions are introduced.