Stability of the weak Haagerup property under graph products

In this paper we prove that: Any graph product of finitely many groups, all of them satisfying weak Haagerup property with $Λ_{WH}=1$, also satisfies weak Haagerup property and as a corollary of this result we obtain that the free product of weakly Haagerup groups with $Λ_{WH}=1$, again has weak Haagerup property with $Λ_{WH}=1$.

💡 Research Summary

The paper investigates the stability of the weak Haagerup property (WH) under graph products of groups. After recalling the hierarchy of approximation properties—amenability, the Haagerup property, weak amenability (with Cowling‑Haagerup constant Λ_CH), and finally the weak Haagerup property (with constant Λ_WH)—the authors focus on groups for which Λ_WH = 1. By definition, a group G has WH if there exists a sequence of functions {ϕ_n} that vanish at infinity, converge pointwise to 1, and have uniformly bounded completely bounded (B₂) norms; the infimum of such bounds is Λ_WH(G). Knudby’s characterization shows that Λ_WH(G)=1 is equivalent to the existence of a proper symmetric function ϕ that can be written as ϕ(y⁻¹x)=‖R(x)−R(y)‖²+‖S(x)+S(y)‖², where the first term yields a conditionally negative definite (CND) kernel and the second a bounded positive definite (PD) kernel.

For each vertex group G_v of a finite simplicial graph Γ, the authors assume the existence of such a decomposition ϕ_v=ρ_v+τ_v with Λ_WH(G_v)=1. Using Schoenberg’s lemma, the CND part ρ_v gives rise to PD kernels e^{‑tρ_v}. Lemma 2.4 provides a Hilbert‑space representation of PD kernels via bounded vector families, and the “exponential” construction Exp⁰ maps vectors into a Fock‑type space where inner products reproduce e^{‑‖·‖²}. This machinery allows the authors to treat both ρ_v and τ_v uniformly as PD kernels with B₂‑norm ≤ 1.

The main technical challenge is to combine the kernels from different vertex groups into a single kernel on the graph product G(Γ). The reduced word description of elements in G(Γ) (due to Green) ensures a well‑defined “reduced length” metric. By summing the CND kernels ρ_v across vertices, one obtains a global CND kernel ρ on G(Γ). For the PD part, the authors embed each τ_v via Exp⁰∘S_v and then take tensor products according to the reduced word structure; the resulting kernel τ remains positive definite and retains B₂‑norm ≤ 1 because tensor products of bounded vectors preserve the supremum norm.

Consequently the function ϕ=ρ+τ on G(Γ) satisfies the requirements for Λ_WH=1: its exponential e^{‑ϕ_n} forms a WH approximate identity. Hence any finite graph product of groups with Λ_WH=1 also has Λ_WH=1. As a special case, when Γ has no edges the graph product reduces to the free product, yielding Corollary 1.2: the free product of two WH groups with Λ_WH=1 again has Λ_WH=1.

The authors note that this result provides further evidence for Cowling’s conjecture (Λ_CH=1 ⇔ Haagerup property), since all known groups with Λ_WH=1 also enjoy the Haagerup property. The paper suggests future work on extending stability to more general amalgamated products and on establishing the converse direction of the conjecture.