  We use subgroup distortion to determine the rate of escape of a simple random
walk on a class of polycyclic groups, and we show that the rate of escape is
invariant under changes of generating set for these groups. For metabelian
groups, we define a stronger form of subgroup distortion, which applies to
non-finitely generated subgroups. Under this hypothesis, we compute the rate of
escape for certain random walks on metabelian groups via a comparison to the
toppling of a dissipative abelian sandpile.
