Uniform almost flatness in finitely generated soluble groups

TL;DR

This paper characterizes finitely generated soluble groups that are virtually nilpotent through uniform polynomial bounds on diameters of finite coset spaces and quotients.

Contribution

It extends previous work by establishing new criteria for virtual nilpotency based on diameter bounds in finitely generated soluble and abelian-by-cyclic groups.

Findings

Finitely generated soluble groups are virtually nilpotent iff their finite coset spaces have polynomial diameter bounds.

Certain abelian-by-cyclic groups also satisfy this condition under weaker diameter assumptions.

The results generalize earlier work by the author and Tointon.

Abstract

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.