Entropy of random walks on groups and the Macaev norm

Abstract

We show that a discrete group on which there is a finitary random walk with positive entropy satisfies a certain condition involving the Macaev norm. This links the entropy of random walks on groups to the author’s work on quasicentral approximate units relative to normed ideals in perturbation theory. On the other hand, the condition we are considering is also an analogue for the Macaev norm of Yamasaki’s hyperbolicity condition.