Virtual endomorphisms of nilpotent groups

Abstract

A virtual endomorphism of a group $G$ is a homomorphism $f:H\to G$ where $H$ is a subgroup of $G$ of finite index $m$. The triple $(G,H,f)$ produces a state-closed (or, self-similar) representation $\phi$ of $G$ on the $1$-rooted $m$-ary tree. This paper is a study of properties of the image $G^{\phi }$ when $G$ is nilpotent. In particular, it is shown that if $G$ is finitely generated, torsion-free and nilpotent then $G^{\phi }$ has solvability degree bounded above by the number of prime divisors of $m$.