Following the work of Burger, Iozzi and Wienhard for representations, in this paper we introduce the notion of maximal measurable cocycles of a surface group. More precisely, let $\mathbf{G}$ be a semisimple algebraic $\mathbb{R}$-group such that $G=\mathbf{G}(\mathbb{R})^\circ$ is of Hermitian type. If $\Gamma \leq L$ is a torsion-free lattice of a finite connected covering of $\text{PU}(1,1)$, given a standard Borel probability $\Gamma$-space $(\Omega,\mu_\Omega)$, we introduce the notion of Toledo invariant for a measurable cocycle $\sigma:\Gamma \times \Omega \rightarrow G$ with an essentially unique boundary map. The Toledo invariant is a multiplicative constant, hence it remains unchanged along $G$-cohomology classes and its absolute value is bounded by the rank of $G$. This allows to define maximal measurable cocycles. We show that the algebraic hull $\mathbf{H}$ of a maximal cocycle $\sigma$ is reductive, the centralizer of $H=\mathbf{H}(\mathbb{R})^\circ$ is compact, $H$ is of tube type and $\sigma$ is cohomologous to a cocycle stabilizing a unique maximal tube-type subdomain. This result is analogous to the one obtained for representations. We conclude with some remarks about boundary maps of maximal Zariski dense cocycles.

中文翻译：

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表面群到厄米李群的最大可测环的代数包

继 Burger、Iozzi 和 Wienhard 对表征的工作之后，在本文中，我们引入了表面群的最大可测共环的概念。更准确地说，令 $\mathbf{G}$ 是一个半单代数 $\mathbb{R}$-群，使得 $G=\mathbf{G}(\mathbb{R})^\circ$ 是 Hermitian 类型。如果$\Gamma \leq L$ 是$\text{PU}(1,1)$ 的有限连通覆盖的无扭格，则给定标准Borel 概率$\Gamma$-space $(\Omega,\ mu_\Omega)$，我们引入了 Toledo 不变量的概念，用于具有本质上唯一的边界图的可测量环圈 $\sigma:\Gamma\times\Omega\rightarrow G$。Toledo 不变量是一个乘法常数，因此它沿 $G$-上同调类保持不变，其绝对值受 $G$ 的等级限制。这允许定义最大可测量的共循环。我们证明了一个最大共环 $\sigma$ 的代数外壳 $\mathbf{H}$ 是约简的，$H=\mathbf{H}(\mathbb{R})^\circ$ 的中心化器是紧凑的，$ H$ 属于管型，$\sigma$ 与稳定唯一最大管型子域的cocycle 同源。这一结果类似于为表征获得的结果。最后，我们对最大 Zariski 密集环的边界图进行了一些评论。