On the Center-Valued Atiyah Conjecture for $L^2$-Betti Numbers

Abstract

The so-called Atiyah conjecture states that the $\mathcal{N}(G)$-dimensions of the $L^2$-homology modules of finite free $G$-CW-complexes belong to a certain set of rational numbers, depending on the finite subgroups of $G$. In this article we extend this conjecture to a statement for the center-valued dimensions. We show that the conjecture is equivalent to a precise description of the structure as a semisimple Artinian ring of the division closure $D(\mathbb{Q}[G])$ of $\mathbb{Q}[G]$ in the ring of affiliated operators. We prove the conjecture for all groups in Linnell's class $\mathfrak{C}$, containing in particular free-by-elementary amenable groups. The center-valued Atiyah conjecture states that the center-valued $L^2$-Betti numbers of finite free $G$-CW-complexes are contained in a certain discrete subset of the center of $\mathbb{C}[G]$, the one generated as an additive group by the center-valued traces of all projections in $\mathbb{C}[H]$, where $H$ runs through the finite subgroups of $G$. Finally, we use the approximation theorem of Knebusch [15] for the center-valued $L^2$-Betti numbers to extend the result to many groups which are residually in $\mathfrak{C}$, in particular for finite extensions of products of free groups and of pure braid groups.