  A combination of Bestvina--Brady Morse theory and an acyclic reflection group
trick produces a torsion-free finitely presented Q-Poincar\'e duality group
which is not the fundamental group of an aspherical closed ANR Q-homology
manifold.
  The acyclic construction suggests asking which Q-Poincar\'e duality groups
act freely on Q-acyclic spaces, i.e., which groups are FH(Q). For example, the
orbifold fundamental group \Gamma\ of a good orbifold satisfies Q-Poincar\'e
duality, and we show \Gamma\ is FH(Q) if the Euler characteristics of certain
fixed sets vanish.
