Weak amenability of hyperbolic groups

Abstract

We prove that hyperbolic groups are weakly amenable. This partially extends the result of Cowling and Haagerup showing that lattices in simple Lie groups of real rank one are weakly amenable. We take a combinatorial approach in the spirit of Haagerup and prove that for the word length distance d of a hyperbolic group, the Schur multipliers associated with the kernel r d have uniformly bounded norms for 0 < r < 1. We then combine this with a Bożejko–Picardello type inequality to obtain weak amenability.