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

Abstract

For bordered surfaces S, we develop a complete parallel between the geometry of the combinatorial Teichm\"uller space $T_S^{comb}$ equipped with Kontsevich symplectic form $\omega_K$, and then the usual Weil-Petersson geometry of Teichm\"uller space $T_S$. The basis for this is an identification of $T_S^{comb}$ with a space of measured foliations with transverse boundary conditions. We equip $T_S^{comb}$ with an analog of the Fenchel-Nielsen coordinates (defined similarly as Dehn-Thurston coordinates) and show they are Darboux for $\omega_K$ (analog of Wolpert formula). We then set up the geometric recursion of Andersen-Borot-Orantin to produce mapping class group invariants functions on $T_S^{comb}$ whose integration with respect to Kontsevich volume form satisfy topological recursion. Further we establish an analog of Mirzakhani-McShane identities, and provide applications to the study of the enumeration of multicurves with respect to combinatorial lengths and Masur-Veech volumes. The formalism allows us to provide uniform and completely geometric proofs of Witten's conjecture/Kontsevich theorem and Norbury's topological recursion for lattice point count in the combinatorial moduli space, parallel to Mirzakhani's proof of her recursion for Weil-Petersson volumes. We strengthen results of Mondello and Do on the convergence of hyperbolic geometry to combinatorial geometry along the rescaling flow, allowing us to flow systematically natural constructions on the usual Teichm\"uller space to their combinatorial analogue, such as a new derivation of the piecewise linear structure of $T_S^{comb}$ originally obtained in the work of Penner, as the limit under the flow of the smooth structure of $T_S$.