Stability of Haagerup property under graph product

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.

💡 Research Summary

The paper investigates the stability of the Haagerup property (also known as a‑T‑menability) under graph products of groups. After a brief historical overview of amenability and the Haagerup property, the authors recall several equivalent characterizations of the Haagerup property: the existence of a proper conditionally negative definite (CND) function, the existence of a proper positive‑definite kernel, and the existence of a proper affine isometric action on a Hilbert space. They note that the property is known to be stable under direct products, free products, and amalgamated products over finite subgroups, and they pose the natural question of whether it is also stable under the more general graph product construction.

A graph product (G(\Gamma)) is defined for a finite simplicial graph (\Gamma) without loops or multiple edges, by taking the free product of the vertex groups ({G_v}_{v\in V(\Gamma)}) and imposing commutation relations (