A cohomological characterisation of Yus Property A for metric spaces

Property A was introduced by Yu as a non-equivariant analogue of amenability. Nigel Higson posed the question of whether there is a homological characterisation of property A. In this paper we answer Higson’s question affirmatively by constructing analogues of group cohomology and bounded cohomology for a metric space X, and show that property A is equivalent to vanishing cohomology. Using these cohomology theories we also give a characterisation of property A in terms of the existence of an asymptotically invariant mean on the space.

💡 Research Summary

The paper answers Nigel Higson’s long‑standing question about a homological description of Yu’s Property A by constructing two cohomology theories for arbitrary metric spaces that play the same role as group cohomology and bounded cohomology do for discrete groups. The authors begin by introducing a controlled cochain complex Cⁿ(X;V) for a metric space X and a Banach X‑module V. A cochain is a function f : Xⁿ⁺¹ → V whose support is uniformly bounded: there exists a constant R such that f(x₀,…,x_n)≠0 only when all pairwise distances d(x_i,x_j)≤R. This “control” condition guarantees that the usual coboundary operator d satisfies d²=0 and that the complex reflects the large‑scale geometry of X.

Two families of coefficient modules are considered. The first is the space ℓ^∞(X) of all bounded real‑valued functions on X, and its closed subspace ℓ^∞₀(X) consisting of functions with zero mean. These modules are natural analogues of the regular and reduced modules used in group cohomology. With ℓ^∞₀(X) as coefficients the authors define the ordinary cohomology Hⁿ(X;ℓ^∞₀) and, using ℓ^∞(X), the bounded cohomology H_bⁿ(X;ℓ^∞).

The central result (Theorem 1) states that a metric space X has Property A if and only if the ordinary cohomology Hⁿ(X;ℓ^∞₀) vanishes for every n≥1. Moreover, Property A implies the vanishing of the bounded cohomology H_bⁿ(X;ℓ^∞) for all n≥1 (Theorem 2). The proof proceeds in two directions.

(i) Property A ⇒ cohomology vanishes.
Property A is known to be equivalent to the existence of an asymptotically invariant mean: a family of probability measures μ_x on X such that ‖μ_x−μ_y‖₁→0 as d(x,y)→∞. The authors use μ to define an averaging operator A on cochains: \