the residual chain are completely open for arbitrary residually finite $G$.

For $G={{\mathbb Z}}^k$ and every field $k$ Elek showed that $\lim _{i\to\infty} b_n(G_i\backslash X;k)$ exists and expresses it in terms of the entropy of $G$-actions on the Pontrjagin duals of finitely generated $kG$-modules [@Elek(2002)] – his techniques play an important role in this paper (see Section \[sec:Eleks\_dimension\_function\]). It was observed in [@Abert-Jaikin-Zapirain-Nikolov(2007)]\*[Theorem 17]{} that the mere convergence of the right hand side of \[the:dim\_approximation\_over\_fields:finitely\_presented\_modules\] in Theorem \[the:dim\_approximation\_over\_fields\] for every amenable $G$ and every field $k$ follows from a general convergence principle for subadditive functions on amenable groups [@Lindenstrauss-Weiss(2000)] and a theorem by Weiss [@Weiss(2001)].

The main purpose of this paper is to determine the limit $\lim _{i\to\infty} b_n(G_i\backslash X;k)$ in algebraic terms for a large class of amenable groups including virtual torsionfree elementary amenable groups. This makes the limit computable by homological techniques; see e.g., the spectral sequence argument of Example \[exa:s1\_actions\].

More precisely, the limit will be expressed in terms of the *Ore dimension*. The group ring $kG$ of a torsionfree amenable group satisfying the zero-divisor conjecture fulfills the Ore condition with respect to the subset $S=kG-\{0\}$ [@Lueck(2002) Example 8.16 on page 324]; we will review the Ore 