曲面群的极大可测上圈到Hermite李群的代数壳
Algebraic hull of maximal measurable cocycles of surface groups into Hermitian Lie groups
摘要：
继Burger、Iozzi和Wienhard关于表示的工作之后，本文引入了曲面群的极大可测上圈的概念。更精确地说，令$\mathbf{G}$是一个半单代数$\mathbb{R}$ -群，使得$G=\mathbf{G}(\mathbb{R})^\circ$是埃尔米特型的。如果$Γ\leq L$是$\text{PU}(1,1)$的有限连通覆盖的无挠格，给定标准Borel概率$Γ$ -空间$(Ω,μ_Ω)$，我们引入了可测上圈$σ:Γ\times Ω\rightarrow G$的Toledo不变量的概念. Toledo沿着$G$ -上同调类保持不变，其绝对值由$G$的秩限制。这允许定义最大可测余圈。证明了极大上循环$σ$的代数壳$\mathbf{H}$是可约的，$H=\mathbf{H}(\mathbb{R})^\circ$的中心化子是紧的.如果另外$σ$承认一个边界图，那么$H$是管型和$σ$是cohomologous一个cocycle稳定一个独特的最大管型子域。这一结果类似于对表象所得到的结果。在特定情况下，$G=\text{PU}(n,1)$极大性足以证明$σ$是上同调的一个上循环保持复杂的测地线。最后，我们给出了极大Zagliki稠密上圈的边界映射。
英文摘要：
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 $Γ\leq L$ is a torsion-free lattice of a finite connected covering of $\text{PU}(1,1)$, given a standard Borel probability $Γ$-space $(Ω,μ_Ω)$, we introduce the notion of Toledo invariant for a measurable cocycle $σ:Γ\times Ω\rightarrow G$. The Toledo 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 $σ$ is reductive and the centralizer of $H=\mathbf{H}(\mathbb{R})^\circ$ is compact. If additionally $σ$ admits a boundary map, then $H$ is of tube type and $σ$ is cohomologous to a cocycle stabilizing a unique maximal tube-type subdomain. This result is analogous to the one obtained for representations. In the particular case $G=\text{PU}(n,1)$ maximality is sufficient to prove that $σ$ is cohomologous to a cocycle preserving a complex geodesic. We conclude with some remarks about boundary maps of maximal Zariski dense cocycles.
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