Bi-exact Wreath-like Product Groups

In this short paper, we prove the bi-exactness of all wreath-like product groups $G\in \mathcal{W}\mathcal{R}(A,B\curvearrowright I)$, whenever $A$ is amenable, $B$ is bi-exact, and the action $B\curvearrowright I$ has amenable stabilizers. As applications, we obtain solidity for the associated group von Neumann algebras and, by combining our result with existing rigidity theorems, new examples exhibiting the McDuff superrigidity.