Decay of concentration functions for adapted probabilities on discrete groups

Given a probability measure, μ , on a locally compact group, necessary and sufficient support conditions which ensure that the concentration functions associated with μ converge to zero have previously been determined. In this note the rate of this convergence when μ is adapted on a discrete group G is shown to depend on the volume growth rate of N _μ , the smallest normal subgroup a coset of which contains the support of μ.