Relative bi-exactness and structural results for graph-wreath product von Neumann algebras

We study relative bi-exactness of graph product and graph-wreath product group von Neumann algebras. In particular, we obtain the relative bi-exactness for graph product von Neumann algebras $LH_{\Gamma}=\ast_{v,\Gamma} LH_v$ and graph-wreath product von Neumann algebras $L(H_{\Gamma}\rtimes G)=(\ast_{v,\Gamma} LH)\rtimes G$, assuming that the component groups are exact. We adopt the $C^{\ast}$-algebraic method of Ozawa for the proof. As an application, for a certain class of graph-wreath products, we establish the rigidity result for the quotient graph $G\backslash\Gamma$ under stable isomorphism. Furthermore, we obtain a new family of prime $\mathrm{II}_1$ factors.