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

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 of any connected surface, possibly except for the Klein bottle, is a retract of each finitely generated group containing as a verbally closed subgroup. We prove that the Klein bottle group is indeed an exception but has a very close property.
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