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arXiv · 1802.09083

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

Rolando de Santiago·Sujan Pant
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Abstract

We demonstrate von Neumann algebra arising from an icc group $Γ$ 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(Γ) $ must arise from direct product decomposition of $Γ$ 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(Γ) $ 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(Γ) $ correspond to a splitting of $Γ$ as a product by groups which are also poly-$\mathcal{C}_\text{rss}$ up to commensurability.

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Rolando de Santiago, Sujan Pant. 2018-02-25. Classification of Tensor Decompositions of II$_1$ Factors Associated With Poly-Hyperbolic Groups. https://arxiv.org/abs/1802.09083

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