The Strong Approximation Conjecture holds for amenable groups

Let G   be a finitely generated group and G▷G1▷G2▷⋯G▷G1▷G2▷⋯ be normal subgroups such that ⋂k=1∞Gk={1}. Let A∈Matd×d(CG)A∈Matd×d(CG) and Ak∈Matd×d(C(G/Gk))Ak∈Matd×d(C(G/Gk)) be the images of A   under the maps induced by the epimorphisms G→G/GkG→G/Gk. According to the strong form of the Approximation Conjecture of Lück [W. Lück, L2L2-Invariants: Theory and Applications to Geometry and K-theory, Ergeb. Math. Grenzgeb. (3), vol. 44, Springer-Verlag, Berlin, 2002]dimG(kerA)=limk→∞dimG/Gk(kerAk), where dimGdimG denotes the von Neumann dimension. In [J. Dodziuk, P. Linnell, V. Mathai, T. Schick, S. Yates, Approximating L2L2-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].