An Analogue of the Torus Decomposition Theorem for Certain Poincaré Duality Groups

Abstract

It is shown that Poincare duality groups which satisfy the maximal condition on centralisers have a canonical decomposition as the fundamental group of a finite graph of groups in which the edge groups are polycyclic-by-finite. The results give useful information only when there are large polycyclic subgroups. Since 3-manifolds groups satisfy Max-c, the results provide a purely group theoretic proof of the Torus Decomposion Theorem. In general, fundamental groups of closed aspherical manifolds satisfy Poincare duality and in fact many of the known examples satisfy Max-c. Thus the results provide a new approach to aspherical manifolds of higher dimensions.