ArXived article on the combinatorial Teichmüller space
Jørgen Ellegaard Andersen and collaborators archives a 105-pages paper entitled: “On the Kontsevich geometry of the combinatorial Teichmüller space”
Professor, Centre Director and D-IAS Chair Jørgen Ellegaard Andersen has together with a group of collaborators from MPIM Bonn, G. Borot, S. Charbonnier, A. Giacchetto, D. Lewanski (also affiliated with IPhT-Saclay and IHES) and C. Wheeler published new research on arXiv entitled “On the Kontsevich geometry of the combinatorial Teichmüller space”.
In the article they study the combinatorial Teichmüller space and construct on it global coordinates, analogous to the Fenchel-Nielsen coordinates on the ordinary Teichmüller space.
They 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. They 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 functions on the combinatorial Teichmüller spaces. They establish a combinatorial analogue of the Mirzakhani-McShane identity fitting this framework.
As applications, they obtain geometric proofs of Witten conjecture/Kontsevich theorem (Virasoro 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 unified theory and proceed in perfect parallel to Mirzakhani's proof of topological recursion for the Weil-Petersson volumes. They move on to the study of the spine construction and the associated rescaling flow on the Teichmüller space. They strengthen former results of Mondello and Do on the convergence of this flow.
In particular, they prove convergence of hyperbolic Fenchel-Nielsen coordinates to the combinatorial ones with some uniformity. This allows them to effectively carry natural constructions on the Teichmüller space to their analogues in the combinatorial spaces. For instance, they obtain the piecewise linear structure on the combinatorial Teichmüller space as the limit of the smooth structure on the Teichmüller space.
To conclude, they provide further applications to the enumerative geometry of multicurves, Masur-Veech volumes and measured foliations in the combinatorial setting.
Link to the paper on arXiv: https://arxiv.org/abs/2010.11806