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Approximating \(L ^{2}\)-invariants and the Atiyah conjecture. (English) Zbl 1036.58017

Let \(G\) be a discrete and torsion-free group, \(K\) a subfield of \(\mathbb{C}\) (complex numbers) and \(A\in M(D\times d,KG)\). One considers \(l^2(G)= \{f:G\to \mathbb{C}/ \sum_{g\in G}| f(g)|^2< \infty\}\) and the bounded linear operator \(A:l^2(G)^d\to l^2(G)^d\) induced by \(A\) by left convolution. Then one says that \(G\) verifies the strong Atiyah conjecture over \(KG\) if \(\dim_G (\ker A)\in \mathbb{Z}\) for every \(A\in M(d\times d,KG)\). The strong Atiyah conjecture over \(KG\) implies that \(KG\) does not contain nontrivial zero divisors.
On the other hand, let \(D\) be the smallest class of groups such that:
(i) If \(G\) is torsion free, \(H\) is elementary amenable (the smallest class of groups containing all cyclic and all finite groups and which is closed under taking group extensions and directed unions), and \(p:G\to H\) is an epimorphism such that \(p^{-1}(E)\in D\) for every finite subgroup \(E\) of \(H\), then \(G\in D\).
(ii) \(D\) is subgroup closed.
(iii) Let \(G_i\in D\) be a directed system of groups and \(G\) its (direct or inverse) limit. Then \(G\in D\).
The authors prove that if \(G\in D\), then the strong Atiyah conjecture over \(\overline{Q}G\) is true, where \(\overline{Q}\) is the field of algebraic numbers in \(\mathbb{C}\). As a consequence they obtain that ther are no nontrivial zero divisors in \(\mathbb{C} G\).
The proof of the above result about the strong Atiyah conjecture is obtained using the main results of the paper which are new approximation properties for \(L^2\)-Betti numbers over \(\overline{Q}G\): Let \(G\) be a group with a sequence of formal subgroups \(G\supset G_1\supset G_2\supset \dots\) such that \(\bigcap_{i\in \mathbb{N}}G_i= \{1\}\) and \(G/G_i\in D\) for every \(i\in \mathbb{N}\). Let \(p_i:G\to G/G_i\) be the natural epimorphism and \(B\in M(d\times d,\overline{Q}G)\). Then, \(\dim_G (\ker(B))= \lim_{i\to \infty} \dim_{G/G_i} \ker(B[i])\), where \(B[i]\) is the image of \(B\) under the ring homomorphism induced by \(p_i\). This result holds for all matrices over \(\mathbb{C} G\) when \(G\) is torsion free and elementary amenable, and \(G/G_i\) belongs to a larger class of groups that contains the class \(D\).

MSC:

58J20 Index theory and related fixed-point theorems on manifolds
57Q10 Simple homotopy type, Whitehead torsion, Reidemeister-Franz torsion, etc.
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
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