Property (T) and Poincar\'e duality in dimension three

TL;DR

This paper proves that residually finite 3-dimensional Poincaré duality groups do not have property (T), leading to new insights about the structure of 3-manifold groups and their geometric properties.

Contribution

It establishes that such groups lack property (T), providing a new proof of Fujiwara's theorem and linking Poincaré duality, property (T), and 3-manifold group properties.

Findings

Residually finite 3D Poincaré duality groups never have property (T)

Groups with property (T) in this class are finite

The proof relies on coboundary expansion results

Abstract

We use a recent result of Bader and Sauer on coboundary expansion to prove residually finite three-dimensional Poincar\'e duality groups never have property (T). This implies such groups are never K\"ahler. The argument applies to fundamental groups of (possibly non-aspherical) compact 3-manifolds, giving a new proof of a theorem of Fujiwara that states if the fundamental group of a compact 3-manifold has property (T), then that group is finite. The only consequence of geometrization needed in the proof is that 3-manifold groups are residually finite.