Preprint 2020-06
Gaetan Borot, Reinier Kramer, and Yannik Schüler
On the Kontsevich Geometry of the Combinatorial Teichmüller Space.
Abstract: We study the combinatorial Teichmüller space and construct on it global coordinates, analogousto the Fenchel–Nielsen coordinates on the ordinary Teichmüller space. We prove that these coordinates form an atlas with piecewise linear transition functions, and constitute global Darboux coordinates for the Kontsevich symplectic structure on top-dimensional cells.
We then set up the geometric recursion in the sense of Andersen–Borot–Orantin adapted to the combinatorial setting, which naturally produces mapping class group invariant func- tions on the combinatorial Teichmu ̈ller spaces. We establish a combinatorial analogue of the Mirzakhani–McShane identity fitting this framework.
As applications, we obtain geometric proofs of Witten conjecture/Kontsevich theorem (Vi- rasoro constraints for ψ-classes intersections) and of Norbury’s topological recursion for the lattice point count in the combinatorial moduli spaces. These proofs arise now as part of a uni- fied theory and proceed in perfect parallel to Mirzakhani’s proof of topological recursion for the Weil–Petersson volumes.
We move on to the study of the spine construction and the associated rescaling flow on the Teichmu ̈ller space. We strengthen former results of Mondello and Do on the convergence of this flow. In particular, we prove convergence of hyperbolic Fenchel–Nielsen coordinates to the combinatorial ones with some uniformity. This allows us to effectively carry natural constructions on the Teichmüller space to their analogues in the combinatorial spaces.Forinstance, we obtain the piecewise linear structure on the combinatorial Teichmüller space as the limit of the smooth structure on the Teichmüller space.
To conclude, we provide further applications to the enumerative geometry of multicurves, Masur–Veech volumes and measured foliations in the combinatorial setting.
Preprint series: Institut für Mathematik, Humboldt-Universität zu Berlin (ISSN 0863-0976), 2020-06
105 pp.