McDuff and Prime von Neumann algebras arising from Thompson-Like Groups

Rolando de Santiago; Patrick DeBonis; Krishnendu Khan

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arXiv · 2312.08345

McDuff and Prime von Neumann algebras arising from Thompson-Like Groups

Rolando de Santiago·Patrick DeBonis·Krishnendu Khan

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Abstract

In this paper we show that the cloning system construction of Skipper and Zaremsky [SZ21], under sufficient conditions, gives rise to Thompson-Like groups which are stable; in particular, these are McDuff groups in the sense of Deprez and Vaes [DV18]. This answers a question of Bashwinger and Zaremsky posed in [BZ23] in the affirmative. In the opposite direction, we show that the group von Neumann algebra for the Higman-Thompson groups $T_d$ and $V_d$ are both prime II$_1$ factors. This follows from a new deformation/rigidity argument for a certain class of groups which admit a proper cocycle into a quasi-regular representation that is not necessarily weakly $\ell^2$.

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Rolando de Santiago, Patrick DeBonis, Krishnendu Khan. 2023-12-13. McDuff and Prime von Neumann algebras arising from Thompson-Like Groups. https://arxiv.org/abs/2312.08345

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