Strong ergodicity, property (T), and orbit equivalence rigidity for translation actions
Strong ergodicity, property (T), and orbit equivalence rigidity for translation actions
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DOI：
10.1515/crelle-2014-0155
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发表时间：
2014-06
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通讯作者：
A. Ioana
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作者：
A. Ioana
我们研究等价关系，产生的平移行动$\Gamma\curvearrowright G$这是相关的密集嵌入$\Gamma<G$的可数群到第二可数局部紧群。假设$G$是单连通的，且作用$\Gamma\curvearrowright G$是强遍历的，我们证明了$\Gamma\curvearrowright G$与另一个平移作用$\Lambda\curvearrowright H$是轨道等价的当且仅当存在同构$\delta：G\rightarrow H$使得$\delta（\Gamma）=\Lambda$.如果$G$此外是一个真实的代数群，那么我们建立类似的刚性结果的平移作用$\Gamma$上的齐次空间的形式$G/\Sigma$，其中$\Sigma<G$是一个离散或代数子群。我们还证明了，如果$G$是单连通的，且作用$\Gamma\curvearrowright G$具有性质（T），则任何上循环$w：\Gamma\times G\rightarrow\Lambda$的值都在可数群$\Lambda$中，则它与同态$\delta：\Gamma\rightarrow\Lambda$上同调.因此，我们推出的行动$\Gamma\curvearrowright G$是轨道等价超刚性：任何自由的非奇异行动$\Lambda\curvearrowright Y$，这是轨道等价于$\Gamma\curvearrowright G$，必然共轭到一个感应$\Gamma\curvearrowright G$。
We study equivalence relations that arise from translation actions $\Gamma\curvearrowright G$ which are associated to dense embeddings $\Gamma<G$ of countable groups into second countable locally compact groups. Assuming that $G$ is simply connected and the action $\Gamma\curvearrowright G$ is strongly ergodic, we prove that $\Gamma\curvearrowright G$ is orbit equivalent to another such translation action $\Lambda\curvearrowright H$ if and only if there exists an isomorphism $\delta:G\rightarrow H$ such that $\delta(\Gamma)=\Lambda$. If $G$ is moreover a real algebraic group, then we establish analogous rigidity results for the translation actions of $\Gamma$ on homogeneous spaces of the form $G/\Sigma$, where $\Sigma<G$ is either a discrete or an algebraic subgroup. We also prove that if $G$ is simply connected and the action $\Gamma\curvearrowright G$ has property (T), then any cocycle $w:\Gamma\times G\rightarrow\Lambda$ with values into a countable group $\Lambda$ is cohomologous to a homomorphism $\delta:\Gamma\rightarrow\Lambda$. As a consequence, we deduce that the action $\Gamma\curvearrowright G$ is orbit equivalent superrigid: any free nonsingular action $\Lambda\curvearrowright Y$ which is orbit equivalent to $\Gamma\curvearrowright G$, is necessarily conjugate to an induction of $\Gamma\curvearrowright G$.