Equidistribution of dense subgroups on nilpotent Lie groups

Let Gamma be a dense subgroup of a simply connected nilpotent Lie group G generated by a finite symmetric set S. We consider the n-ball S-n for the word metric induced by S on Gamma. We show that S-n (with uniform measure) becomes equidistributed on G with respect to the Haar measure as n tends to infinity. We also prove the analogous result for random walk averages.