Polycyclic groups are virtually poly-infinite cyclic
= Polycyclic groups are virtually poly-infinite cyclic
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>.