  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.
