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Pré-Publication, Document De Travail Année : 2021

Around the combinatorial unit ball of measured foliations on bordered surfaces

Résumé

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.

Dates et versions

hal-03424338 , version 1 (10-11-2021)

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Gaëtan Borot, Séverin Charbonnier, Vincent Delecroix, Alessandro Giacchetto, Campbell Wheeler. Around the combinatorial unit ball of measured foliations on bordered surfaces. 2021. ⟨hal-03424338⟩
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