Uniform almost flatness in finitely generated soluble groups

We show that a finitely generated soluble group is virtually nilpotent if and only if the diameter of its finite coset spaces admits a uniform polynomial lower bound in terms of their size. We obtain the same conclusion for certain finitely generated abelian-by-cyclic groups under the weaker assumption that the diameters of their finite quotients are uniformly bounded below by a polynomial in their size. This extends the previous work of the author with Tointon.