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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 groupsuch that $L(\oplus_{n\in \mathbb N} \Gamma_n)\cong L(\Lambda)$ then thereexists 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 rigidityproperty found in [CdSS16]. Using this we provide an uncountable family ofrestricted wreath products $\Gamma\cong\Sigma\wr \Delta$ of icc, property (T)groups $\Sigma$, $\Delta$ whose wreath product structure is recognizable, up toa normal amenable subgroup, from their von Neumann algebras $L(\Gamma)$. Alongthe way we highlight several applications of these results to the study ofrigidity in the $\mathbb C^*$-algebra setting.

Some rigidity results for II$_1$ factors arising from wreath products of property (T) groups