Finiteness properties for some rational Poincaré duality groups (Q384303)

Davis asked whether every torsion--free finitely presented \(R\)--Poincaré duality group is the fundamental group of an aspherical closed \(R\)--homology manifold. Following Davis, the author applies a combination of Bestvina--Brady Morse theory and an acyclic reflection group trick to answer the question in the negative by constructing a counterexample for \(R = \mathbb Q\). The construction leads to the question which groups are fundamental groups of finite complexes with \(R\)--acyclic universal cover, that is, which groups satisfy property FH(\(R\)) introduced by Bestvina and Brady. In the context of orbifolds this amounts to a finiteness obstruction and the paper describes an equivariant finiteness theory similar to that of Lück and Davis, using modules over categories. ``The orbifold fundamental group of a good orbifold satisfies \(\mathbb Q\)--Poincaré duality'' and the main result shows that such a group satisfies FH(\(\mathbb Q\)) ``if the Euler characteristic of certain fixed sets vanish.'' Another application of the reflection group trick and lattices with torsion provide further examples.