Finiteness properties for some rational Poincaré duality groups

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

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