Let G be a torsion-free discrete group, and let Q denote the field of algebraic numbers in C. We prove that QG fulfills the Atiyah conjecture if G lies in a certain class of groups D, which contains in particular all groups that are residually torsion-free elementary amenable or are residually free. This result implies that there are no nontrivial zero divisors in CG. The statement relies on new approximation results for L 2 -Betti numbers over QG, which are the core of the work done in this paper. Another set of results in the paper is concerned with certain number-theoretic properties of eigenvalues for the combinatorial Laplacian on L 2 -cochains on any normal covering space of a finite CW complex. We establish the absence of eigenvalues that are transcendental numbers whenever the covering transformation group is either amenable or in the Linnell class C. We also establish the absence of eigenvalues that are Liouville transcendental numbers whenever the covering transformation group is either residually finite or more generally in a certain large bootstrap class G.

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If G is an elementary amenable group of finite Hirsch length h , then the quotient of G by its maximal locally finite normal subgroup has a maximal solvable normal subgroup, of derived length and index bounded in terms of h .

The Atiyah conjecture for a discrete group G states that the L 2 -Betti numbers of a finite CW-complex with fundamental group G are integers if G is torsion-free, and in general that they are rational numbers with denominators determined by the finite subgroups of G.Here we establish conditions under which the Atiyah conjecture for a torsion-free group G implies the Atiyah conjecture for every finite extension of G. The most important requirement is that H * (G, Z/p) is isomorphic to the cohomology of the p-adic completion of G for every prime number p. An additional assumption is necessary, e.g. that the quotients of the lower central series or of the derived series are torsion-free.We prove that these conditions are fulfilled for a certain class of groups, which contains in particular Artin's pure braid groups (and more generally fundamental groups of fiber-type arrangements), free groups, fundamental groups of orientable compact surfaces, certain knot and link groups, certain primitive one-relator groups, and products of these. Therefore every finite, in fact every elementary amenable extension of these groups satisfies the Atiyah conjecture, provided the group does.As a consequence, if such an extension H is torsion-free then the group ring CH contains no non-trivial zero divisors, i.e. H fulfills the zero-divisor conjecture.In the course of the proof we prove that if these extensions are torsionfree, then they have plenty of non-trivial torsion-free quotients which are virtually nilpotent. All of this applies in particular to Artin's full braid group, therefore answering question B6 on www.grouptheory.info.Our methods also apply to the Baum-Connes conjecture. This is discussed in [46], where for example the Baum-Connes conjecture is proved for the full braid groups.MSC: 55N25 (homology with local coefficients), 16S34 (group rings, Laurent rings), 57M25 (knots and links)

Abstract. We first prove that the Whitehead group of a torsion-free virtually solvable linear group vanishes. Next we make a reduction of the fibered isomorphism conjecture from virtually solvable groups to a class of virtually solvable Q-linear groups. Finally we prove an L-theory analogue for elementary amenable groups.