  In this article, we examine how the structure of soluble groups of infinite
torsion-free rank with no section isomorphic to the wreath product of two
infinite cyclic groups can be analysed. As a corollary, we obtain that if a
finitely generated soluble group has a defined Krull dimension and has no
sections isomorphic to the wreath product of two infinite cyclic groups then it
is a group of finite torsion-free rank. There are further corollaries including
applications to return probabilities for random walks. The paper concludes with
constructions of examples that can be compared with recent constructions of
Brieussel and Zheng.
