The strong approximation conjecture holds for amenable groups

Let G be a finitely generated group and G (sic) G(1) (sic) G(2) (sic)... be normal subgroups such that boolean AND(infinity)(k=1) G(k) = {1}. Let A is an element of Mat(dxd)(CG) and A(k) is an element of Mat(dxd)(C(G/G(k))) be the images of A under the maps induced by the epimorphisms G --> G/G(k). According to the strong form of the Approximation Conjecture of Luck [W. Luck, L-2-Invariants: Theory and Applications to Geometry and K-theory, Ergeb. Math. Grenzgeb. (3), vol. 44, Springer-Verlag, Berlin, 2002] dim(G)(ker A)= lim(k-->infinity) dim(G/Gk)(ker A(k)), where dim(G) denotes the von Neumann dimension. In [J. Dodziuk, P. Linnell, V. Mathai, T Schick, S. Yates, Approximating L-2-invariants and the Atiyah conjecture, Comm. Pure Appl. Math. 56 (7) (2003) 839-873] Dodziuk et al. proved the conjecture for torsion free elementary amenable groups. In this paper we extend their result for all amenable groups, using the quasi-tilings of Ornstein and Weiss [D.S. Ornstein, B. Weiss, Entropy and isomorphism theorems for actions of amenable groups, J. Anal. Math. 48 (1987) 1-141]. (C) 2005 Elsevier Inc. All rights reserved.