arXiv · 1703.04254
In this paper, we propose a new approach to Cwikel estimates both for the Euclidean space and for the noncommutative Euclidean space.
Galina Levitina, Fedor Sukochev, Dmitriy Zanin. 2017-03-13. Cwikel estimates revisited. https://doi.org/10.1112/plms.12301
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We show that von Neumann algebras of fundamental groups of closed orientable surfaces of genus $g\geq2$ are free group factors on $2g-1$generators. The key technical ingredient involves a proof that the element $w=ABA^{-1}B^{-1}$ of the free group $\mathbb{F}_{2}=\langle A,B\rangle$ is freely complemented in the group factor: $L(\mathbb{F}_{2})=W^{*}(w)*W^{*}(v)$ for some Haar unitary $v\in L(\mathbb{F}_{2})$ that is freely independent from $w$. Combined with previous results, we conclude that for an arbitrary finitely generated torsion-free non-elementary discrete subgroup $\Gamma\subset PSL_{2}(\mathbb{R})$, $L(\Gamma)$ is a free group factor, settling a conjecture of de la Harpe and Voiculescu. This result was obtained using OpenAI's ChatGPT Pro 6.0.
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