Around the combinatorial unit ball of measured foliations on bordered surfaces

The volume \(B^{\mathrm{comb}}_{\Sigma}(\mathbb{G})\) 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 \((B^{\mathrm{comb}}_{\Sigma})^s\), as a function over the combinatorial moduli spaces, is integrable with respect to the Kontsevich measure. The results depends on the topology of \(\Sigma\), in contrast with the situation for hyperbolic surfaces where Arana-Herrera and Athreya recently proved an optimal square-integrability.