[Paper Review] Hyperbolic metrics, measured foliations and pants decompositions for non-orientable surfaces

Athanase Papadopoulos, Robert Penner|arXiv (Cornell University)|Sep 2, 2013

Geometric and Algebraic Topology15 references17 citations

TL;DR

This paper establishes analogues of the Fenchel-Nielsen, Dehn-Thurston, and Hatcher-Thurston theorems for non-orientable surfaces, introducing modified Fenchel-Nielsen coordinates that exclude twisting numbers for 1-sided curves, adapting Dehn-Thurston coordinates by omitting twisting for 1-sided curves, and extending the Hatcher-Thurston move set with two new moves to account for 1-sided curves. The key contribution is a complete, minimal parametrization of Teichmüller space and measured foliations, and a compactification of Teichmüller space via a piecewise-linear sphere of projective measured foliations of compact support.

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.

Motivation & Objective

• To extend foundational theorems of Teichmüller theory—previously valid only for orientable surfaces—to non-orientable surfaces.
• To resolve the absence of systematic parametrizations of Teichmüller space and measured foliations for non-orientable surfaces in the literature.
• To introduce a generalized Fenchel-Nielsen theorem that omits twisting parameters for 1-sided curves in pants decompositions.
• To provide a generalized Dehn-Thurston theorem that excludes twisting numbers for 1-sided curves.
• To extend the Hatcher-Thurston move set with two new moves to ensure transitivity on pants decompositions of non-orientable surfaces.

Proposed method

• Adapts Fenchel-Nielsen coordinates by using only hyperbolic lengths and twisting numbers for 2-sided curves in a pants decomposition of a non-orientable surface.
• Introduces a new definition of twisting numbers that depends on curve orientation and basepoints, generalizing the orientable case.
• Modifies the Dehn-Thurston parametrization by omitting intersection and twisting numbers for curves that represent punctures or 1-sided curves.
• Extends the Hatcher-Thurston move set with two new elementary moves: one replacing a 1-sided curve with another 1-sided curve, and another replacing two 1-sided curves with a 2-sided curve or vice versa.
• Constructs a compactification of Teichmüller space using a piecewise-linear sphere of projective measured foliations of compact support.
• Uses local charts near boundary points via equidistant foliations to define a homeomorphism between measured foliations and a subset of the space of measured foliations transverse to a pants decomposition.

Experimental results

Research questions

• RQ1 How can the Fenchel-Nielsen theorem be generalized to non-orientable surfaces, particularly in the absence of twisting numbers for 1-sided curves?
• RQ2 What is the correct parametrization of measured foliations on non-orientable surfaces, and how does it differ from the Dehn-Thurston theorem in the orientable case?
• RQ3 What additional moves are required to ensure that the set of pants decompositions of a non-orientable surface is connected under elementary moves, beyond the classical Hatcher-Thurston moves?
• RQ4 How is Thurston's boundary of Teichmüller space realized for non-orientable surfaces, and what is the topological structure of the space of projective measured foliations of compact support?
• RQ5 What is the dimension of the sphere that compactifies Teichmüller space for a non-orientable surface, and how does it depend on the number of 1-sided and 2-sided curves in a pants decomposition?

Key findings

• The generalized Fenchel-Nielsen theorem provides a parametrization of Teichmüller space for non-orientable surfaces using only hyperbolic lengths of curves in a pants decomposition and twisting numbers for 2-sided curves, omitting twisting for 1-sided curves.
• The generalized Dehn-Thurston theorem parametrizes measured foliations using intersection numbers and twisting numbers, with both omitted for 1-sided curves and curves representing punctures.
• The generalized Hatcher-Thurston theorem adds two new elementary moves to the classical two, ensuring that the set of all pants decompositions of a non-orientable surface is connected under these moves.
• Thurston's boundary for the Teichmüller space of a non-orientable surface is realized as a piecewise-linear sphere of dimension $\#{\mathcal{P}}_1 + 2\#{\mathcal{P}}_2 - 1$, where $\#{\mathcal{P}}_1$ and $\#{\mathcal{P}}_2$ are the numbers of 1-sided and 2-sided curves in a pants decomposition.
• The space of projective measured foliations of compact support, denoted $\mathcal{PF}_0(S)$, forms a closed piecewise-linear ball that compactifies Teichmüller space into a closed ball, with the mapping class group acting continuously on this boundary.