Howson groups which are not strongly Howson

TL;DR

This paper constructs the first examples of Howson groups that are not strongly Howson, demonstrating that the property of being Howson does not imply the stronger uniform bound condition.

Contribution

It provides the first explicit examples of Howson groups lacking the strongly Howson property, clarifying the relationship between these classes.

Findings

Constructed the first examples of Howson groups that are not strongly Howson.

Showed that the class of Howson groups is strictly larger than the class of strongly Howson groups.

Abstract

A group $G$ is called a Howson group if the intersection $H\cap K$ of any two finitely generated subgroups $H, K<G$ is again finitely generated, and called a strongly Howson group when a uniform bound for the rank of $H\cap K$ can be obtained from the ranks of $H$ and $K$. Clearly, every strongly Howson group is a Howson group, but it is unclear in the literature whether the converse is true. In this note, we show that the converse is not true by constructing the first Howson groups which are not strongly Howson.