Structure of normally and finitely non-co-Hopfian groups

TL;DR

This paper investigates the structure of finitely generated groups that contain infinite descending chains of proper normal finite-index subgroups isomorphic to the group itself, revealing their connection to free abelian quotients and applications to scale-invariant groups.

Contribution

It characterizes finitely generated groups with self-similar chains of subgroups, linking them to free abelian quotients and addressing conjectures on scale-invariant groups.

Findings

Such groups are, up to finite index, pullbacks from free abelian quotients.

Characteristic finite-index subgroups isomorphic to G come from the abelianization.

Provides partial results supporting conjectures on scale-invariant 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 characteristic proper finite-index subgroup isomorphic to G arises by pulling back a finite-index subgroup of the abelianization, and secondly, we prove special cases (for normal subgroups) of conjectures of Benjamini and Nekrashevych--Pete regarding the classification of scale-invariant groups.