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

Soluble groups with no $\mathbb{Z} \wr \mathbb{Z}$ sections