Structure of normally and finitely non-co-Hopfian groups
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Publication:6292213
DOI10.4171/GGD/603arXiv1710.02179MaRDI QIDQ6292213
Publication date: 5 October 2017
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.
Other classes of groups defined by subgroup chains (20F22) Chains and lattices of subgroups, subnormal subgroups (20E15) Limits, profinite groups (20E18)
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