Structure of normally and finitely non-co-Hopfian groups

Abstract

A group $G$ is (finitely) co-Hopfian if it does not contain any proper (finite-index) subgroups isomorphic to itself. We study finitely generated groups $G$ that admit a descending chain of proper normal finite-index subgroups, each of which is isomorphic to $G$. We prove that up to finite index, these are always obtained by pulling back a chain of subgroups from a free abelian quotient. We give two applications: first, we show any proper self-embedding of $G$ with finite-index characteristic image arises by pulling back an endomorphism of the abelianization; secondly, we prove special cases (for normal subgroups) of conjectures of Benjamini and Nekrashevych–Pete regarding the classification of scale-invariant groups.