The Strong Approximation Conjecture holds for amenable groups

∞k =1Gk = {1}. Let A ∈ Matd×d (CG) and Ak ∈ Matd×d (C(G/Gk)) be the images of A under the maps induced by the epimorphisms G→G/Gk. According to the strong form of the Approximation Conjecture of Lück [W. Lück, L2-Invariants: Theory and Applications to Geometry and K-theory, Ergeb. Math. Grenzgeb. (3), vol. 44, Springer-Verlag, Berlin, 2002] dimG(ker A) = lim k→∞ dimG/Gk (kerAk), where dimG denotes the von Neumann dimension. In [J. Dodziuk, P. Linnell, V. Mathai, T. Schick, S. Yates, Approximating L2-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]. © 2005 Elsevier Inc. All rights reserved.