| Résume | I will describe several results regarding the statistics of multicurves on bordered surfaces, whose combinatorial lengths are bounded by a cut-off parameter.
After a description of the combinatorial Teichmüller spaces, I will first state how such statistics can be computed by geometric recursion, a recursive procedure akin to topological recursion. Second, the asymptotics of the number of multicurves as the cut-off tends to infinity allow to define a function on combinatorial Teichmüller spaces, that is interpreted as the volume of the combinatorial unit ball of measured foliations; it is the combinatorial analogue of Mirzakhani's B function in the hyperbolic setup. It descends to the moduli spaces and the structure of the latter allows to completely determine its range of integrability with respect to the Kontsevich measure. The range shows surprising dependence on the topology of the surface. Along the talk, I will compare the results with those holding in the hyperbolic world.
Joint works with J. E. Andersen, G. Borot, V. Delecroix, A. Giacchetto, D. Lewański and C. Wheeler. |
