Strictly ascending HNN extensions in soluble groups

TL;DR

This paper constructs examples of finitely generated soluble groups that are not LERF but lack strictly ascending HNN extensions, solving a problem in the Kourovka notebook and expanding understanding of group extension properties.

Contribution

It demonstrates the existence of soluble groups with specific extension properties, providing counterexamples to previously open questions.

Findings

Existence of finitely generated soluble groups not LERF without strictly ascending HNN extensions

Construction of a finitely presented soluble non-LERF group without such extensions of polycyclic groups

Resolution of Problem 16.2 in the Kourovka notebook

Abstract

We show that there exist finitely generated soluble groups which are not LERF but which do not contain strictly ascending HNN extensions of a cyclic group. This solves Problem 16.2 in the Kourovka notebook. We further show that there is a finitely presented soluble group which is not LERF but which does not contain a strictly ascending HNN extension of a polycyclic group.