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Intersection norms on surfaces and Birkhoff sections for geodesic flows

Marcos Cossarini and Pierre Dehornoy
Algebraic & Geometric Topology 25 (2025) 4499–4545
DOI: 10.2140/agt.2025.25.4499
Abstract

Every filling multicurve on a smooth surface determines a norm on the first homology group of the surface. The unit ball of the dual norm is the convex hull of finitely many integer points. We give an interpretation of these points in terms of certain coorientations of the multicurve. Our main result is a classification statement: when the surface is hyperbolic and the filling multicurve is geodesic, integer points in the interior of the unit ball of the dual norm classify isotopy classes of Birkhoff sections for the geodesic flow (on the unit tangent bundle to the surface) whose boundary is the symmetric lift of the multicurve. All results remain true when one replaces the hyperbolic surface by a 2-dimensional orientable hyperbolic orbifold.

Keywords
curve, surface, geodesic, fibered knot, Birkhoff section
Mathematical Subject Classification
Primary: 37D40
Secondary: 37D45, 57K30, 57N37
References
Publication
Received: 21 September 2020
Revised: 19 September 2024
Accepted: 13 October 2024
Published: 20 November 2025
Authors
Marcos Cossarini
Laboratoire d’analyse et de mathématiques appliquées
Université Paris-Est Créteil
Créteil
France
Pierre Dehornoy
Institut de Mathématiques de Marseille
Aix-Marseille Université
Marseille
France
https://www.i2m.univ-amu.fr/perso/pierre.dehornoy
© 2025 The Author(s), under exclusive license to MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY).

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