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It seems to be known that Hirsch length and cohomological dimension agree for (torsion-free, finitely generated) polycyclic groups.

If we drop the assumption "torsion-free", then cd is of course infinite. But, is it still true (as one might expect) that the rational cohomological dimension is bounded above by the Hirsch length?

More generally, are there known conditions on a group G such that cd(G)≥cd(H) if there is a surjective homomorphism G-->H? (For the Hirsch length this inequality is immediate from the definition.)

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Regarding the parenthetical "finitely generated": all polycyclic groups are finitely generated. – Stephen S Oct 11 '11 at 8:16
Doesn't the question on rational cohomological dimension follow directly from the Hochschild-Serre s.s.? – Torsten Ekedahl Oct 11 '11 at 9:27
up vote 5 down vote accepted

Hillman extended the notion of Hirsch length to elementary amenable groups and proved that it is bounded above by the rational cohomological dimension. The reference is

Jonathan A. Hillman, Elementary amenable groups and 4-manifolds with Euler characteristic 0, J. Austral. Math. Soc. (Series A) 50 (1991), 160-170.

To answer the question in the comments about nilpotent groups, theorem 5, section 8.8 in Gruenberg's book "Cohomological topics in group theory" (Springer LNM 143) says that for a torsion free nilpotent group $G$ with finite Hirsch length, one has $cd(G)=h(G)$ when $G$ is finitely generated and $cd(G)=h(G)+1$ otherwise.

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What about nilpotent groups? Does one have equality in this case? – anonymous Oct 11 '11 at 10:03

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