Timeline for Additive integer-valued functions on the module category

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Jun 21, 2012 at 9:28	comment	added	Andreas Thom		Atiyah asked in the last 70s if $\ell^2$-Betti numbers are always rational. This question has turned into a conjecture over the years, with various modifications for torsionfree groups and groups with bounded torsion. Finally, some have been disproved by Grigorchuk-Zuk, and later Austin, Grabowski and Schick-Zuk (in various papers). The torsionfree case is still open. If true, then Lück's map above is integer-valued. Lück defined the map in his study of dimension-functions in the 90's, also Elek has a definition of rank of modules over the group ring of an amenable group.
Jun 20, 2012 at 9:37	comment	added	Ralph		Thanks for this very interesting answer. A comment and a question: 1) The log-example can also be used if $R$ is finite (i.e. $\lambda(M)=\log\|M\|$), giving some variant of the length. 2) In your explanation you relate the Atiyah conjecture and Lück's map $\varphi$. From a historical point of view, was the Atiyah conjecture first ?
Jun 20, 2012 at 8:34	history	edited	Andreas Thom	CC BY-SA 3.0	added 7 characters in body
Jun 20, 2012 at 8:29	history	edited	Andreas Thom	CC BY-SA 3.0	added 1150 characters in body; deleted 4 characters in body; added 142 characters in body
Jun 20, 2012 at 8:19	history	answered	Andreas Thom	CC BY-SA 3.0