The volume $\mathscr{ B}_{ \Sigma}^{ { \rm comb}}(\mathbb{ G})$ 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 $(\mathscr{ B}_{ \Sigma}^{ { \rm comb}})^{ s}$, 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 (arXiv:1907.06287) recently proved an optimal square-integrability.
Categories: math.GT math.CO math.DG