The volume $\mathcal {B}_{\Sigma }^{\textrm {comb}}({\mathbb {G}})$ 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 $(\mathcal {B}_{\Sigma }^{\textrm {comb}})^{s}$, as a function over the combinatorial moduli spaces, is integrable with respect to the Kontsevich measure. The results depend on the topology of $\Sigma $, in contrast with the situation for hyperbolic surfaces where [6] recently proved an optimal square integrability.

中文翻译：

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在边界表面上测量的叶片组合单元球周围

单位球的体积 $\mathcal {B}_{\Sigma }^{\textrm {comb}}({\mathbb {G}})$——关于组合长度函数 $\ell _{{\ mathbb {G}}}$——在稳定边界表面 $\Sigma $ 上测量的叶面空间作为 $\Sigma $ 上多曲线数量的多项式增长的前因数出现。我们找到 $s \in {\mathbb {R}}$ 的范围，其中 $(\mathcal {B}_{\Sigma }^{\textrm {comb}})^{s}$, 作为一个函数组合模空间，对于 Kontsevich 测度是可积的。结果取决于 $\Sigma $ 的拓扑结构，与双曲曲面的情况相反，其中 [6] 最近证明了最佳平方可积性。