Property (T) and Poincaré duality in dimension three

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.

💡 Research Summary

The paper establishes that a residually finite three‑dimensional Poincaré duality (PD³) group cannot have Kazhdan’s Property (T). The authors achieve this by combining three recent developments: (i) Bader and Sauer’s coboundary expansion theorem for groups of type FP₂ with Property (T), (ii) the algebraic formulation of Poincaré duality for groups, and (iii) a non‑expansion result for residually finite groups.

Main Result (Theorem 1).
If G is a residually finite PD³ group, then G does not have Property (T). The proof proceeds by contradiction. Assuming G has Property (T), Bader–Sauer’s theorem (stated as Theorem 3 in the paper) provides a uniform constant C such that for every finite‑index normal subgroup H⊲G and every coboundary η∈C²(F|H;ℤ) there exists a cochain ω∈C¹(F|H;ℤ) with dω=η and ‖ω‖≤C‖η‖. Here F→ℤ is any based partial resolution of length two (type FL₂).

Because G is a PD³ group, its cohomology with coefficients in ℤG is concentrated in degree 3, and the duality isomorphism H³(G;ℤG)≅ℤ yields a “codimension‑two linear isoperimetric inequality” (Proposition 1.11). In concrete terms, for each finite‑index subgroup H the 2‑dimensional cellular complex X/H (obtained from a suitable 2‑complex X on which G acts freely and cocompactly) satisfies a linear lower bound |∂A| ≥ c|A| for any 2‑chain A filling a non‑trivial boundary ∂A. The constant c>0 depends only on G.

The authors then invoke a new non‑expansion theorem (Theorem 4). For any residually finite infinite group G with a residual chain {G_i}, either some finite cover has positive first Betti number, or for every ε>0 there exists a non‑trivial boundary z in some cover M_i such that any filling 2‑chain A satisfies |z| < ε|A|. In particular, for a PD³ group with trivial first Betti number in all finite covers, one can make the boundary‑to‑area ratio arbitrarily small.

These two statements are incompatible: the linear isoperimetric inequality forces a uniform positive lower bound on |∂A|/|A|, while Theorem 4 guarantees the existence of arbitrarily small ratios. Hence the assumption that G has Property (T) leads to a contradiction, proving Theorem 1.

Corollary 1.
A residually finite PD³ group cannot be a Kähler group. This follows because any PD³ Kähler group must have Property (T) (by unpublished work of Delzant, or by theorems of Kotschick combined with Reznikov). Consequently, the class of residually finite PD³ groups is disjoint from the class of Kähler groups.

Application to 3‑Manifold Groups (Theorem 2).
The same argument applies to the fundamental group of any compact 3‑manifold, regardless of asphericity. If π₁(M) had Property (T), then it would be a residually finite PD³ group (or contain such a subgroup), contradicting Theorem 1. Therefore π₁(M) must be finite. This recovers Fujiwara’s theorem, but the proof only uses the fact that 3‑manifold groups are residually finite—a consequence of modern work on virtual specialness (Agol, Wise) and does not require the full geometrization theorem.

Technical Framework.
The paper begins by recalling the algebraic set‑up for group cohomology with normed ℤG‑modules, defining partial resolutions of type FPₙ and FLₙ, and establishing that the ℓ¹‑norm on cochains is independent of the choice of coset representatives (Proposition 1.2). It then presents the Bader–Sauer results in the FP₂ context (Theorems 1.3–1.5) and shows how to pass from real to integral coefficients (Theorem 1.6). A key step is Lemma 1.8, which proves that the ℤ‑expansion property is invariant under changing the partial resolution, using chain homotopy arguments reminiscent of Gersten’s work.

The authors define PDⁿ groups in the usual algebraic sense (finite projective resolution and dualizing module ℤ), discuss orientable versus non‑orientable cases, and note that any non‑orientable PDⁿ group contains an index‑two orientable PDⁿ subgroup.

Finally, the paper situates its results within broader conjectures: Cannon’s conjecture (hyperbolic groups with S² boundary are lattices in SO⁺(3,1)), the existence of non‑residually finite PDⁿ groups for n>3, and the relationship between weaker forms of Property (τ) and coboundary expansion. The authors suggest that their method may illuminate the “Lubotzky–Sarnak conjecture” in dimension three, as PD³ groups cannot satisfy a stronger version of Property (τ) that would imply uniform coboundary expansion.

Overall Significance.
The work provides a purely algebraic and cohomological proof that residually finite PD³ groups lack Property (T), thereby giving a new, streamlined proof of Fujiwara’s finiteness theorem for 3‑manifold groups. It showcases how recent advances in higher‑dimensional coboundary expansion can be leveraged to address classical problems in low‑dimensional topology, reducing reliance on deep geometric inputs such as the full geometrization theorem. The techniques introduced may have further applications to the study of Property (τ), expanders, and the topology of groups beyond dimension three.