The Strong Approximation Conjecture holds for amenable groups

Abstract

Let G be a finitely generated group and G ▷ G 1 ▷ G 2 ▷ ⋯ be normal subgroups such that ⋂ k = 1 ∞ G k = { 1 } . Let A ∈ Mat d × d ( C G ) and A k ∈ Mat d × d ( 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 Lück [W. Lück, 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 → ∞ dim G / G k ( 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].