Around the combinatorial unit ball of measured foliations on bordered surfaces

TL;DR

This paper investigates the integrability properties of the volume of the combinatorial unit ball of measured foliations on bordered surfaces, revealing dependence on surface topology and contrasting with hyperbolic surface results.

Contribution

It determines the range of exponents for which the volume function's powers are integrable over combinatorial moduli spaces, highlighting topological influences.

Findings

Identifies integrability conditions depending on surface topology.

Shows contrast with hyperbolic surface case where optimal square-integrability is established.

Provides new insights into the volume growth of measured foliations on bordered surfaces.

Abstract

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.