  We study the return probability of finitely generated metabelian groups. We
give a characterization of metabelian groups whose return probability is
equivalent to $\exp(-n^\frac1{3} )$ in purely algebraic terms, namely the Krull
dimension of the group. Along the way, we give lower bounds on the return
probability for metabelian groups with torsion derived subgroup, according to
the dimension. We also establish a variation of the famous embedding theorem of
Kaloujinine and Krasner for metabelian groups that respects the Krull
dimension.
