Finiteness properties for some rational Poincar\'e duality groups

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.