Finiteness properties

A combination of Bestvina–Brady Morse theory and an acyclic reflection group trick produces a torsion-free finitely presented \mathbb{Q} -Poincaré duality group which is not the fundamental group of an aspherical closed ANR \mathbb{Q} -homology manifold.

The acyclic construction suggests asking which \mathbb{Q} -Poincaré duality groups act freely on \mathbb{Q} -acyclic spaces (i.e., which groups are \mbox{FH}(\mathbb{Q}) ). For example, the orbifold fundamental group \Gamma of a good orbifold satisfies \mathbb{Q} -Poincaré duality, and we show \Gamma is \mbox{FH}(\mathbb{Q}) if the Euler characteristics of certain fixed sets vanish.

Download as PDF