Property $\mathrm{(TTT)}$ and the Cowling--Haagerup constant

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