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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 ₖ₇ (G) is strictly greater than 1. We apply this result to some semidirect products and lattices in higher rank algebraic groups.
I. Vergara (Sat,) studied this question.