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 $L{H}_{\mathrm{\Gamma }}={\ast }_{v,\mathrm{\Gamma }}L{H}_v$ and graph-wreath product von Neumann algebras $L(H_{\mathrm{\Gamma }}⋊G)=({\ast }_{v,\mathrm{\Gamma }}LH)⋊G$, 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 $G\mathrm{\backslash }\mathrm{\Gamma }$ under stable isomorphism. Furthermore, we obtain a new family of prime ${\mathrm{I}\mathrm{I}}_1$ factors.