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

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.