Structure of normally and finitely non-co-Hopfian groups

2017/10/05 by van Limbeek, Wouter · 3 citations
#FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.1710.02179

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.

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