Geodesic
On the Center-Valued Atiyah Conjecture for $L^2$-Betti Numbers
Documenta mathematica, Tome 22 (2017), pp. 659-677
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The so-called Atiyah conjecture states that the N(G)-dimensions of the L2-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(Q[G]) of Q[G] in the ring of affiliated operators. We prove the conjecture for all groups in Linnell's class C, containing in particular free-by-elementary amenable groups. The center-valued Atiyah conjecture states that the center-valued L2-Betti numbers of finite free G-CW-complexes are contained in a certain discrete subset of the center of C[G], the one generated as an additive group by the center-valued traces of all projections in C[H], where H runs through the finite subgroups of G. Finally, we use the approximation theorem of Knebusch [15] for the center-valued L2-Betti numbers to extend the result to many groups which are residually in C, in particular for finite extensions of products of free groups and of pure braid groups.
DOI : 10.4171/dm/575
Classification : 20C07, 46L10, 46L80, 47A58
Mots-clés : Atiyah conjecture, center-valued trace, von Neumann dimension, L2-Betti numbers 
@article{10_4171_dm_575,
     author = {Anselm Knebusch and Peter Linnell and Thomas Schick},
     title = {On the {Center-Valued} {Atiyah} {Conjecture} for $L^2${-Betti} {Numbers}},
     journal = {Documenta mathematica},
     pages = {659--677},
     year = {2017},
     volume = {22},
     doi = {10.4171/dm/575},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/dm/575/}
}
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Anselm Knebusch; Peter Linnell; Thomas Schick. On the Center-Valued Atiyah Conjecture for $L^2$-Betti Numbers. Documenta mathematica, Tome 22 (2017), pp. 659-677. doi: 10.4171/dm/575

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