On the Haagerup inequality and groups acting on Ã

Let Gamma be a group endowed with a length function L, and let E be a linear subspace of C Gamma. We say that E satisfies the Haagerup inequality if there exists constants C, s > 0 such that, for any f is an element of E, the convolutor norm of f on l(2)(Gamma) is dominated by C times the l(2) norm of f(l + L)(s). We show that, for E = C Gamma, the Haagerup inequality can be expressed in terms of decay of random walks associated with finitely supported symmetric probability measures on Gamma. If L is a word length function on a finitely generated group Gamma, we show that, if the space Rad(L)(Gamma) of radial functions with respect to L satisfies the Haagerup inequality, then Gamma is non-amenable if and only if Gamma has superpolynomial growth. We also show that the Haagerup inequality for Rad(L)(Gamma) has a purely combinatorial interpretation; thus, using the main result of the companion paper by J. Swiatkowski, we deduce that, for a group Gamma acting simply transitively on the vertices of a thick euclidean building of type (A) over tilde(n), the space Rad(L)(Gamma) satisfies the Haagerup inequality, and Gamma is non-amenable.