The Kontsevich geometry of the combinatorial Teichmüller spaces
In the early ’80s, a combinatorial description of the moduli spaces of curves was discovered, which led to remarkable results about its topology. Inspired by analogue arguments in the hyperbolic setting, we describe the combinatorial Teichmüller space parametrising marked metric ribbon graphs on a surface. We introduce global Fenchel–Nielsen coordinates and show a Wolpert-type formula for the Kontsevich symplectic form. As applications of this set-up, we present a combinatorial analogue of Mirzakhani’s identity, resulting in a new proof of Witten–Kontsevich recursion for the symplectic volumes of the combinatorial moduli space, and Norbury’s recursion for the counting of integral points. The talk is based on a joint work in progress with J.E. Andersen, G. Borot, S. Charbonnier, D. Lewański and C. Wheeler.
Alessandro Giacchetto, Max Planck Institute for Mathematics, Bonn