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Congruence subgroups and the Atiyah conjecture

Daniel R. Farkas·Peter A. Linnell
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Abstract

Let A denote the algebraic closure of the rationals Q in the complex numbers C. Suppose G is a torsion-free group which contains a congruence subgroup as a normal subgroup of finite index and denote by U(G) the C-algebra of closed densely defined unbounded operators affiliated to the group von Neumann algebra. We prove that there exists a division ring D(G) such that A[G] < D(G) < U(G). This establishes some versions of the Atiyah conjecture for the group G.

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Daniel R. Farkas, Peter A. Linnell. 2006-03-17. Congruence subgroups and the Atiyah conjecture. https://arxiv.org/abs/math/0511747

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