Let $\Gamma$ be a group endowed with a length function $L$, and let $E$ be a linear subspace of $\mathbf{C}\Gamma$. We say that $E$ satisfies the Haagerup inequality if there exists constants $C,s\>0$ such that, for any $f\in E$, the convolutor norm of $f$ on ${\ell }^2\left(\Gamma \right)$ is dominated by $C$ times the ${\ell }^2$ norm of $f(1+L{)}^s$. We show that, for $E=\mathbf{C}\Gamma$, 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,...

We give an estimate for the number of closed loops of given length in the 1-skeleton of a thick euclidean building. This kind of estimate can be used to prove the (RD) property for the subspace of radial functions on ${\tilde{A}}_n$ groups, as shown in the paper by A. Valette [same issue].