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 ${\mathbf{\Lambda }}_{\mathrm{WH}}(G)$ is strictly greater than $1$. We apply this result to some semidirect products and lattices in higher rank algebraic groups.

8 pages. Added new section on higher rank algebraic groups with improved results. Many references added. To appear in Complex Analysis and Operator Theory

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• Non commutative Lp spaces without the completely bounded approximation property
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• The weak Haagerup property
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