On the Kontsevich geometry of the combinatorial Teichm\"uller space

We study the combinatorial Teichmuller space and construct on it global coordinates, analogous to the Fenchel-Nielsen coordinates on the ordinary Teichmuller 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 functions on the combinatorial Teichmuller 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 (Virasoro constraints for $\psi$-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. We move on to the study of the spine construction and the associated rescaling flow on the Teichmuller 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 Teichmuller space to their analogues in the combinatorial spaces. For instance, we obtain the piecewise linear structure on the combinatorial Teichmuller space as the limit of the smooth structure on the Teichmuller space. To conclude, we provide further applications to the enumerative geometry of multicurves, Masur-Veech volumes and measured foliations in the combinatorial setting.