For a compact surface $S = S_{g,n}$ with $3g + n \geq 4$, we introduce a family of unitary representations of the mapping class group $\operatorname{Mod}(S)$ based on the space of measured foliations. or this family of representations, we show that none of them has almost invariant vectors. As one application, we obtain an inequality concerning the action of $\operatorname{Mod}(S)$ on the Teichmüller space of $S$. Moreover, using the same method plus recent results about weak equivalence, we also give a classification, up to weak equivalence, for the unitary quasi-representations with respect to geometrical subgroups.

中文翻译：

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关于映射类群的酉表示族

对于具有 $3g + n \geq 4$ 的紧凑表面 $S = S_{g,n}$，我们引入了基于空间的映射类群 $\operatorname{Mod}(S)$ 的酉表示族测量的叶面。或者这一系列表示，我们表明它们都没有几乎不变的向量。作为一个应用，我们获得了关于 $\operatorname{Mod}(S)$ 在 $S$ 的 Teichmüller 空间上的作用的不等式。此外，使用相同的方法加上最近关于弱等价的结果，我们还对几何子群的幺正拟表示进行了分类，直到弱等价。