Weak amenability of Coxeter groups

TL;DR

This paper demonstrates that Coxeter groups are weakly amenable by extending certain unitary representations to the open unit disc and showing their uniform boundedness.

Contribution

It introduces a novel method of extending positive definite functions to prove weak amenability of Coxeter groups.

Findings

Coxeter groups are weakly amenable.

Constructed uniformly bounded representations extend to the open unit disc.

The approach links positive definite functions to group properties.

Abstract

Let (G,S) be a Coxeter group. We construct a continuation, to the open unit disc, of the unitary representations associated to the positive definite functions g -> r^l(g). (Here 0<r<1, and l denotes the length function with respect to the generating set S.) The constructed representations are uniformly bounded and we prove that this implies the weak amenability of the group G.