We show that any infinite collection $(\Gamma_n)_{n\in \mathbb N}$ of icc, hyperbolic, property (T) groups satisfies the following von Neumann algebraic \emph{infinite product rigidity} phenomenon. If $\Lambda$ is an arbitrary group such that $L(\oplus_{n\in \mathbb N} \Gamma_n)\cong L(\Lambda)$ then there exists an infinite direct sum decomposition $\Lambda=(\oplus_{n \in \mathbb N} \Lambda_n )\oplus A$ with $A$ icc amenable such that, for all $n\in \mathbb N$, up to amplifications, we have $L(\Gamma_n) \cong L(\Lambda_n)$ and $L(\oplus_{k\geq n} \Gamma_k )\cong L((\oplus_{k\geq n} \Lambda_k) \oplus A)$. The result is sharp and complements the previous finite product rigidity property found in [CdSS16]. Using this we provide an uncountable family of restricted wreath products $\Gamma\cong\Sigma\wr \Delta$ of icc, property (T) groups $\Sigma$, $\Delta$ whose wreath product structure is recognizable, up to a normal amenable subgroup, from their von Neumann algebras $L(\Gamma)$. Along the way we highlight several applications of these results to the study of rigidity in the $\mathbb C^*$-algebra setting.

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由属性 (T) 组的花环积引起的 II1 因子的一些刚性结果

我们证明了 icc、双曲、性质 (T) 群的任何无限集合 $(\Gamma_n)_{n\in \mathbb N}$ 满足以下冯诺依曼代数 \emph {无限积刚性} 现象。如果 $\Lambda$ 是一个任意群，使得 $L(\oplus_{n\in \mathbb N} \Gamma_n)\cong L(\Lambda)$ 那么存在无限直和分解 $\Lambda=(\oplus_ {n \in \mathbb N} \Lambda_n )\oplus A$ 和 $A$ icc 适合这样，对于所有 $n\in \mathbb N$，直到放大，我们有 $L(\Gamma_n) \cong L (\Lambda_n)$ 和 $L(\oplus_{k\geq n} \Gamma_k )\cong L((\oplus_{k\geq n} \Lambda_k) \oplus A)$。结果很清晰，补充了之前在 [CdSS16] 中发现的有限产品刚性属性。使用这个，我们提供了不可数的限制花圈产品族 $\Gamma\cong\Sigma\wr \Delta$ 的 icc，属性（T）组 $\Sigma$，从他们的冯诺依曼代数 $L(\Gamma)$ 可以识别出 $\Delta$ 的花环积结构，直到一个正常的顺从子群。在此过程中，我们重点介绍了这些结果在 $\mathbb C^*$-代数设置中刚性研究的几种应用。