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.
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