A group is (finitely) co-Hopfian if it does not contain any proper (finite-index) subgroups isomorphic to itself. We study finitely generated groups that admit a descending chain of proper normal finite-index subgroups, each of which is isomorphic to . 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 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.
Cite this article
Wouter van Limbeek, Structure of normally and finitely non-co-Hopfian groups. Groups Geom. Dyn. 15 (2021), no. 2, pp. 465–489DOI 10.4171/GGD/603