On the Howson property of descending HNN-extensions of groups

TL;DR

This paper investigates the Howson property in descending HNN-extensions of groups, showing that under certain conditions, these extensions do not preserve the property, especially for non-cyclic free groups.

Contribution

It proves that descending HNN-extensions generally lack the Howson property under specific assumptions, extending previous results and clarifying the behavior for free groups.

Findings

Descending HNN-extensions are not Howson groups under certain conditions.

Non-cyclic free groups' descending HNN-extensions lack the Howson property.

The results connect with and extend classical findings by Burns and Brunner.

Abstract

A group $G$ is said to have the Howson property (or to be a Howson group) if the intersection of any two finitely generated subgroups of $G$ is finitely generated subgroup. It is proved that descending HNN-extension is not a Howson group under some assumptions satisfied by the base group of HNN-extension. In particular, a result of the paper joined with a Burns - Brunner result (received in 1979) implies that any descending HNN-extension of non-cyclic free group does not have the Howson property.