Residually finite groups with uniformly almost flat quotients

TL;DR

This paper proves that certain residually finite groups with uniformly almost flat finite quotients are virtually nilpotent, extending previous results to broader classes of groups under polynomial diameter bounds.

Contribution

It establishes new conditions under which residually finite groups are virtually nilpotent, generalizing prior work to include residually torsion-free nilpotent groups.

Findings

Groups with polynomially bounded diameters of finite quotients are virtually nilpotent.

Extends previous results to residually torsion-free nilpotent groups.

Provides a broader criterion for virtual nilpotency in residually finite groups.

Abstract

We show that if all the finite coset spaces of a polycyclic group have diameter bounded uniformly below by a polynomial in their size then the group is virtually nilpotent. We obtain the same conclusion for a finitely generated residually torsion-free nilpotent group under the weaker assumption that the finite quotient groups have diameter bounded uniformly below by a polynomial in their size. This extends work of Khukhro and Valette.