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.)