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Stability of the weak Haagerup property under graph products

Shubhabrata Das, Partha Sarathi Ghosh

TL;DR

The paper proves that graph products preserve the weak Haagerup property when each vertex group has Λ_{WH}=1, showing that the graph product G(Γ) also has WH with Λ_{WH}=1. The authors build a global approximate identity by gluing vertex-level WH data using a CAT(0) cube complex model and a PD-scaling approach, controlling the B_2-norm via PD kernels and Mizuta-type estimates. They derive invariant kernels ψ_{n,Γ} on G(Γ) by carefully analyzing reduced forms and tail contributions, and finally obtain φ_{n,Γ} with the required vanishing-at-infinity and norm-bounded properties. This yields Λ_{WH}(G(Γ))=1 and implies the same stability for free products, providing evidence in favor of the Cowling-like conjecture linking Λ_{WH}=1 to Haagerup-type properties.

Abstract

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 $Λ_{WH}=1$.

Stability of the weak Haagerup property under graph products

TL;DR

The paper proves that graph products preserve the weak Haagerup property when each vertex group has Λ_{WH}=1, showing that the graph product G(Γ) also has WH with Λ_{WH}=1. The authors build a global approximate identity by gluing vertex-level WH data using a CAT(0) cube complex model and a PD-scaling approach, controlling the B_2-norm via PD kernels and Mizuta-type estimates. They derive invariant kernels ψ_{n,Γ} on G(Γ) by carefully analyzing reduced forms and tail contributions, and finally obtain φ_{n,Γ} with the required vanishing-at-infinity and norm-bounded properties. This yields Λ_{WH}(G(Γ))=1 and implies the same stability for free products, providing evidence in favor of the Cowling-like conjecture linking Λ_{WH}=1 to Haagerup-type properties.

Abstract

In this paper we prove that: Any graph product of finitely many groups, all of them satisfying weak Haagerup property with , also satisfies weak Haagerup property and as a corollary of this result we obtain that the free product of weakly Haagerup groups with , again has weak Haagerup property with .
Paper Structure (15 sections, 10 theorems, 79 equations, 6 figures)

This paper contains 15 sections, 10 theorems, 79 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Completely Bounded Maps and $B_2$-norm
  4. Weak Haagerup Property
  5. Exponential of a Hilbert Space
  6. Graph Product of Groups
  7. Construction of Cube Complex from Graph Products
  8. Wall Space structure on Graph Products
  9. Weak Haagerup property for Graph Product
  10. Construction of kernels
  11. $B_2$-norm estimate of $\psi_{n,\Gamma}$
  12. The kernel $\psi_{n,\Gamma}$ is $G(\Gamma)$-invariant
  13. Expression of $f(\eta^{-1}\gamma,1_{G(\Gamma)})$
  14. Expression of $f(\gamma,\eta)$
  15. Proof of the main theorem

Key Result

Theorem 1.1

Suppose $\Gamma$ is a finite simplicial graph and $\{G_{v}\}_{v\in V(\Gamma)}$ is a collection of weakly Haagerup groups with $\Lambda_{WH}(G_v)=1$, for each $v$. Then the group $G(\Gamma)$ has the weak Haagerup property with $\Lambda_{WH}(G(\Gamma))=1$.

Figures (6)

  • Figure 1: $\Gamma$ is disconnected $G(\Gamma)=*_{i=1}^5G_i$
  • Figure 2: $\Gamma$ is complete $G(\Gamma)=\oplus_{i=1}^5G_i$
  • Figure 3: $G_v,G_{v_1},G_{v_2},G_{v_3}$ generates $G(\Gamma_{st(v)})$ in $G(\Gamma)$
  • Figure 4: Schematic diagram of reduced form of $\eta^{-1}\gamma$
  • Figure 5:
  • ...and 1 more figures

Theorems & Definitions (21)

  • Theorem 1.1
  • Corollary 1.2
  • Remark 2.1
  • Lemma 2.2
  • Definition 2.3
  • Lemma 2.4: ozawa2008weak, Theorem 3
  • Definition 2.5
  • Remark 2.6
  • Remark 2.7
  • Definition 2.8
  • ...and 11 more