On the Center-Valued Atiyah Conjecture for -Betti Numbers
Anselm Knebusch
HFT-Stuttgart, GermanyPeter Linnell
Virginia Tech Blacksburg, USAThomas Schick
Mathematisches Institut, Universität Göttingen, Germany
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Abstract
The so-called Atiyah conjecture states that the -dimensions of the -homology modules of finite free -CW-complexes belong to a certain set of rational numbers, depending on the finite subgroups of . 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 of in the ring of affiliated operators. We prove the conjecture for all groups in Linnell's class , containing in particular free-by-elementary amenable groups. The center-valued Atiyah conjecture states that the center-valued -Betti numbers of finite free -CW-complexes are contained in a certain discrete subset of the center of , the one generated as an additive group by the center-valued traces of all projections in , where runs through the finite subgroups of . Finally, we use the approximation theorem of Knebusch [15] for the center-valued -Betti numbers to extend the result to many groups which are residually in , in particular for finite extensions of products of free groups and of pure braid groups.
Cite this article
Anselm Knebusch, Peter Linnell, Thomas Schick, On the Center-Valued Atiyah Conjecture for -Betti Numbers. Doc. Math. 22 (2017), pp. 659–677
DOI 10.4171/DM/575