Stability of the weak Haagerup property under graph products

TL;DR

This paper proves that the weak Haagerup property with a specific constant is preserved under graph and free product constructions of groups, extending the class of groups known to have this property.

Contribution

It establishes the stability of the weak Haagerup property with constant 1 under graph and free product operations, a new result in group property theory.

Findings

Weak Haagerup property with $ extLambda_{WH}=1$ is preserved under graph products.

Free products of weakly Haagerup groups with $ extLambda_{WH}=1$ also have the property.

The result broadens understanding of the stability of the weak Haagerup property.

Abstract

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