Stability of Haagerup property under graph product

TL;DR

The paper proves that the graph product $G(\Gamma)$ of finitely many Haagerup groups $\{G_v\}_{v\in V(\Gamma)}$ has the Haagerup property. It builds a proper conditionally negative definite function on $G(\Gamma)$ by gluing the vertex-group CND data through a CAT(0) cube complex framework, defining a representation $R$ into a Hilbert space and a kernel $k(g,h)=\|R(g)-R(h)\|^2$ that is $G(\Gamma)$-invariant. The resulting CND function $\phi_\Gamma=l_r+\tilde{\phi}$ (where $l_r$ is Haagerup’s reduced length) is proper, giving a Haagerup tuple for $G(\Gamma)$. This establishes that finite graph products preserve the Haagerup property, aligning with and providing a more direct construction than previous approaches. The work reinforces the stability of Haagerup groups under generalized product operations and contributes a constructive method for obtaining proper CND functions on graph products.

Abstract

In this paper, we prove that any graph product of finitely many groups, all satisfying the Haagerup property (or Gromov's a-T-menability) also satisfies Haagerup property.

Figures (2)

• Figure 1: $\Gamma$ is disconnected $G(\Gamma)=*_{i=1}^5G_i$
• Figure 2: $\Gamma$ is complete $G(\Gamma)=\oplus_{i=1}^5G_i$

Theorems & Definitions (15)

• Theorem 1.1
• Lemma 2.1
• Definition 2.2
• Definition 2.3
• Definition 3.1
• Remark 3.2
• Remark 3.3
• Lemma 4.1
• proof
• Lemma 4.2