Finiteness properties for some rational Poincar\'e duality groups

TL;DR

This paper constructs specific torsion-free Q-Poincaré duality groups using Morse theory and reflection groups, and explores their actions on Q-acyclic spaces, revealing new properties of these groups.

Contribution

It introduces a novel construction of torsion-free Q-Poincaré duality groups that are not fundamental groups of aspherical manifolds, and investigates conditions for these groups to act freely on Q-acyclic spaces.

Findings

Constructed a torsion-free finitely presented Q-Poincaré duality group.

Identified conditions under which orbifold fundamental groups are FH(Q).

Showed certain fixed set Euler characteristics vanish for these groups.

Abstract

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.