Groups and Dynamics

We introduce the notion of the Rapid Decay Property (RDP) for Fell bundles over locally compact étale groupoids, and as a main consequence, we resolve the question of localizability: the compatibility of algebra and topology in forming norm completions. We then show that Fell bundles over groupoids of polynomial growth satisfies RDP. Moreover, for Deaconu–Renault groupoids realized as partial crossed products of free groups, the existence of persistent branching forces exponential growth, so RDP fails even though the acting free group itself satisfies it: sharply separating the group and groupoid theories.