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

TL;DR

This paper proves a rigidity phenomenon for tensor decompositions of II$_1$ factors arising from poly-hyperbolic groups, linking algebraic decompositions to group structure and providing criteria for primeness.

Contribution

It establishes that tensor decompositions of certain II$_1$ factors correspond to direct product decompositions of the underlying groups, advancing the understanding of von Neumann algebra rigidity.

Findings

Tensor decompositions correspond to group product decompositions.

L(mma) is prime iff amma is indecomposable as a non-amenable group.

Decompositions of finite index subalgebras relate to group splittings up to commensurability.

Abstract

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.