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

TL;DR

This paper investigates the structure of certain soluble groups lacking specific wreath product sections, revealing conditions under which such groups have finite torsion-free rank and exploring applications to random walks.

Contribution

It provides new structural results for soluble groups without $bZ times bZ$ sections and connects these findings to torsion-free rank and random walk behavior.

Findings

Finitely generated soluble groups with Krull dimension and no wreath product sections have finite torsion-free rank.

Constructs examples of such groups for comparison with recent work.

Applications to return probabilities in random walks on these groups.

Abstract

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.