The Klein bottle group is not strongly verbally closed, though awfully close to being so

TL;DR

This paper investigates the Klein bottle group’s properties related to being verbally closed within finitely generated groups, showing it is not strongly verbally closed but exhibits a closely related property.

Contribution

It proves that the Klein bottle group is an exception to Mazhuga's theorem and introduces a new property that closely resembles being strongly verbally closed.

Findings

Klein bottle group is not strongly verbally closed.

Klein bottle group has a property very close to being strongly verbally closed.

Contrasts with the behavior of fundamental groups of other surfaces.

Abstract

According to Mazhuga's theorem, the fundamental group $H$ of any connected surface, possibly except for the Klein bottle, is a retract of each finitely generated group containing $H$ as a verbally closed subgroup. We prove that the Klein bottle group is indeed an exception but has a very close property.