Every polycyclic group has a finite-index subgroup with a series whose nontrivial factors are infinite cyclic. This is the polycyclic finite-index lemma; it also supplies torsion-free finite-index subgroups. Such a subgroup is a poly-infinite cyclic group. Combining its series with the extension rule for Poincare duality groups makes it a duality group of dimension its Hirsch length.
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