Weak amenability is stable under graph products

TL;DR

This paper proves that weak amenability, an important property in group theory, remains stable when constructing graph products of discrete groups, extending known stability results.

Contribution

It demonstrates that graph products of weakly amenable groups are weakly amenable, providing new methods for extending functions and constructing wall spaces.

Findings

Weak amenability is preserved under graph products.

Constructed a wall space related to graph product word length.

Developed a method to extend completely bounded functions.

Abstract

Weak amenability of discrete groups was introduced by Haagerup and co-authors in the 1980's. It is an approximation property known to be stable under direct products and free products. In this paper we show that graph products of weakly amenable discrete groups are weakly amenable (with Cowling-Haagerup constant 1). Along the way we construct a wall space associated to the word length structure of a graph product and also give a method for extending completely bounded functions on discrete groups to a completely bounded function on their graph product.