Classification of Tensor Decompositions of II$_1$ Factors Associated With Poly-Hyperbolic Groups

We demonstrate von Neumann algebra arising from an icc group $\Gamma$ in Chifan's, Ioana's, and Kida's class of poly-$\mathcal{C}_\text{rss}$, such as a poly-hyperbolic group with no amenable factors in its composition series, satisfies the following rigidity phenomenon discovered in DHI16 (see also CdSS17): every tensor decomposition of the II$_1$ factor $L(\Gamma)$ must arise from direct product decomposition of $\Gamma$ by groups which are poly-$\mathcal{C}_\text{rss}$. Through heavy usage and developments of the techniques in CdSS15, we improve the second author's and their collaborator's work in CKP14 by providing group-level criteria for determining whether a group von Neumann algebra is prime: $L(\Gamma)$ is prime precisely when the group is indecomposable as a direct product of non-amenable groups. We further demonstrate that all tensor decompositions of finite index subalgebras of $L(\Gamma)$ correspond to a splitting of $\Gamma$ as a product by groups which are also poly-$\mathcal{C}_\text{rss}$ up to commensurability.