Virtual endomorphisms of the group $pg

• Published in: Researches in mathematics (Online) Vol. 32; no. 1; pp. 3 - 15
• Main Authors: Bondarenko, I., Zashkolny, D.
• Format: Journal Article
• Language: English
• Published: Oles Honchar Dnipro National University 08.07.2024
• Subjects: plane group; self-similar action; virtual endomorphism
• ISSN: 2664-4991; 2664-5009
• DOI: 10.15421/242401

A virtual endomorphism of a group $G$ is a homomorphism of the form $\phi:H\rightarrow G$, where $H<G$ is a subgroup of finite index. A virtual endomorphism $\phi:H\rightarrow G$ is called simple if there are no nontrivial normal $\phi$-invariant subgroups, that is, the $\phi$-core is trivial. We describe all virtual endomorphisms of the plane group $pg$, also known as the fundamental group of the Klein bottle. We determine which of these virtual endomorphisms are simple, and apply these results to the self-similar actions of the group. We prove that the group $pg$ admits a transitive self-similar (as well as finite-state) action of degree $d$ if and only if $d\geq 2$ is not an odd prime, and admits a self-replicating action of degree $d$ if and only if $d\geq 6$ is not a prime or a power of $2$.