Virtual endomorphisms of the group $pg$

Abstract

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 $2$.