论文标题
表面基团的最大可测量共伴生物群体的代数船体进入冬宫谎言基团
Algebraic hull of maximal measurable cocycles of surface groups into Hermitian Lie groups
论文作者
论文摘要
在汉堡，IOZZI和WIENHARD的作品中进行表示，在本文中，我们介绍了表面基团的最大可测量共生概念。更准确地说，让$ \ mathbf {g} $为半imple代数$ \ mathbb {r} $ - 组，以便$ g = \ mathbf {g}（\ mathbb {r}）^\ circ $ circ $ circ $属于Hermitian类型。如果$γ\ leq l $是$ \ text {pu}（1,1）$的有限连接覆盖物的无扭矩的晶格，则给定标准的borel概率$γ$ -Space $（ω，μ_Ω）$，我们引入了toledo Invareiant of Musurable coccocarear $ cocy $ forde $ f the tosurable $ qucy $ f \ fort。 托莱多在$ g $ o-g $的亚属学课程中保持不变，其绝对价值受$ g $的排名。这允许定义最大可测量的共生。我们表明，最大共生$σ$的代数船体$ \ MATHBF {H} $是还原的，并且$ H = \ Mathbf {h}的centralizer（\ Mathbb {r}）^\ circ $是紧凑的。如果另外$σ$承认边界图，则$ h $是管类型，$σ$对于稳定唯一的最大管型子域的共生稳定。该结果类似于获得的表示。 在特定情况下，$ g = \ text {pu}（n，1）$最大性足以证明$σ$对于保留复杂地理素的共同体是共生的。 我们以关于最大Zariski致密共生的边界图的一些评论。
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.
