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

Athanase Papadopoulos (1, 2) , Robert C. Penner (2, 3, 4)

1 IRMA - Institut de Recherche Mathématique Avancée
2 Erwin Schrödinger Institute for Mathematical Physics
3 Center for the Topology and Quantization of Moduli Spaces
4 Caltech, Mathematics and Physics Departments

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 Teichmüller 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.

Mots clés: Hatcher-Thurston Theorem; non-orientable surfaces; pants decompositions; Teichmü ller space; Thurston's boundary; Fenchel-Nielsen Theorem; Dehn-Thurston Theorem; Hatcher-Thurston Theorem.

Domaines: Topologie géométrique [math.GT]

Athanase Papadopoulos, Robert C. Penner. Hyperbolic metrics, measured foliations and pants decompositions for non-orientable surfaces. Asian Journal of Mathematics, 2016, 20 (1), p. 157-182. ⟨10.4310/AJM.2016.v20.n1.a7⟩. ⟨hal-00856638v3⟩