Property (T) and Poincaré duality in dimension three

TL;DR

The paper shows that residually finite $\mathsf{PD}_3$ groups cannot have property (T), by combining Bader–Sauer's coboundary expansion with Poincaré duality to transfer 2-coboundary data into codimension-two expansion and then to a hyperbolicity obstruction. This yields that such groups are not Kähler and provides a new, geometry-driven route to Fujiwara’s theorem for 3-manifold groups. The approach suggests a broader strategy: ruling out property (T) via higher coboundary expansion and duality, reducing (T) questions to hyperbolicity and its incompatibilities with expansion in low codimensions. Overall, the work connects cohomological expansion, duality, and geometric group theory to derive concrete finiteness results for 3-manifold groups with (T).

Abstract

We use a recent result of Bader and Sauer on coboundary expansion to prove residually finite three-dimensional Poincaré duality groups never have property (T). This implies such groups are never Kähler. 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.

Theorems & Definitions (40)

• Theorem 1
• Corollary 1
• Theorem 2: Fujiwara
• Theorem 3: ${\mathsf{FP}}_2$ version of Theorem 2.13 BaderSauer
• Theorem 4
• Lemma 1.1
• Theorem 1.2: ${\mathsf{FP}}_2$ version of Theorem 1.6 in BaderSauer
• proof
• Theorem 1.3: ${\mathsf{FP}}_2$ version of Theorem 1.7 in BaderSauer
• Theorem 1.4: ${\mathsf{FP}}_2$ version of Theorem 1.8 BaderSauer