Property (Ttt) and the Cowling–Haagerup Constant

We show that Property (TTT) is an obstruction to weak amenability with Cowling–Haagerup constant 1. More precisely, if G is a countable group and H is an infinite subgroup of G such that the pair (G, H) has relative Property (TTT), then the weak Haagerup constant Λ<inf>WH</inf>(G) is strictly greater than 1. We apply this result to some semidirect products and lattices in higher rank algebraic groups. © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2024.