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 $$$2$$$.