Structure of normally and finitely non-co-Hopfian 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.
Keywords
Cite
@article{arxiv.1710.02179,
title = {Structure of normally and finitely non-co-Hopfian groups},
author = {Wouter van Limbeek},
journal= {arXiv preprint arXiv:1710.02179},
year = {2020}
}
Comments
minor corrections to have been made. To appear in Groups Geom. Dyn