Around the Combinatorial Unit Ball of Measured Foliations on Bordered Surfaces

Abstract

Abstract The volume $\mathcal {B}_{\Sigma }^{\textrm {comb}}({\mathbb {G}})$ of the unit ball—with respect to the combinatorial length function $\ell _{{\mathbb {G}}}$—of the space of measured foliations on a stable bordered surface $\Sigma $ appears as the prefactor of the polynomial growth of the number of multicurves on $\Sigma $. We find the range of $s \in {\mathbb {R}}$ for which $(\mathcal {B}_{\Sigma }^{\textrm {comb}})^{s}$, as a function over the combinatorial moduli spaces, is integrable with respect to the Kontsevich measure. The results depend on the topology of $\Sigma $, in contrast with the situation for hyperbolic surfaces where [6] recently proved an optimal square integrability.