Property \(\mathrm {(TTT)}\) and the Cowling–Haagerup Constant

We show that Property \(\mathrm {(TTT)}\) $(\mathrm{TTT})$ 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 \(\mathrm {(TTT)}\) $(\mathrm{TTT})$ , then the weak Haagerup constant \(\varvec{\Lambda }_{\textrm{WH}}(G)\) ${\mathbf{\Lambda }}_{\text{WH}}(G)$ is strictly greater than 1. We apply this result to some semidirect products and lattices in higher rank algebraic groups.