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Let $G$ be a Noetherian group. Is $H_n(G,\mathbb{Z})$ finitely generated? Do we know the above for the special cases $n=2,3$ even?

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    $\begingroup$ For virtually polycyclic this should be true, using the Hochschild-Serre spectral sequence. I guess it should also be true whenever $\mathbb{Z}[G]$ is noetherian (though I do not know if this is strictly more general than virtually polycyclic); there is a paper of S.I. Ivanov "Group rings of noetherian groups" that may be of interest. So the interesting examples would be things like Tarski monster groups, see the book "Geometry of defining relations in groups" by A.Yu. Ol'shanskii. Maybe the methods in there are sufficiently geometric to work out some examples? $\endgroup$ Aug 19, 2014 at 12:46

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