Around the combinatorial unit ball of measured foliations on bordered surfaces

Abstract

The volume ${\mathcal{B}}_{\Sigma }^{\mathrm{comb}}(\mathbb{G})$ 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 $({\mathcal{B}}_{\Sigma }^{\mathrm{comb}}{)}^s$, 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.

37 pages, 2 appendices. v2: typos corrected, pictures added, and explanations added in various places

References in corpus (1)

• Masur-Veech volumes, frequencies of simple closed geodesics and intersection numbers of moduli spaces of curves