Virtual fibring of Poincar\'e-duality groups

TL;DR

This paper characterizes when RFRS Poincaré-duality groups virtually fiber over the integers with Poincaré-duality kernels, linking this to the vanishing of their $L^2$-homology and exploring cohomology at infinity.

Contribution

It establishes a criterion connecting $L^2$-homology vanishing to virtual fibering of Poincaré-duality groups and relates cohomology at infinity to algebraic fibering of group extensions.

Findings

RFRS Poincaré-duality groups admit a virtual epimorphism to $bZ$ with Poincaré-duality kernel iff their $L^2$-homology vanishes.

Cohomology at infinity of algebraically fibering groups relates to their fibers being one-ended.

Aspherical manifold groups that algebraically fiber have fibers with one end.

Abstract

We show that a RFRS Poincar\'e-duality group $G$ admits a virtual epimorphism to the integers whose kernel is itself a Poincar\'e-duality group over every field if and only if the $L^2$-homology of $G$ vanishes and so do the positive-characteristic variants thereof. Our investigations yield a more general relationship between cohomology at infinity of groups that algebraically fibre and their fibres. In particular, we show that if the fundamental group of an aspherical manifold of dimension at least three algebraically fibres, then the fibre is one ended.