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Locally compact modules over abelian groups and compactly generated metabelian groups
Part of:
Locally compact groups and their algebras
Theory of modules and ideals
Chain conditions, finiteness conditions
Locally compact abelian groups
Published online by Cambridge University Press: 27 August 2025
Abstract
We perform a general study of the structure of locally compact modules over compactly generated abelian groups. We obtain a dévissage result for such modules of the form ‘compact-by-sheer-by-discrete’, and then study more specifically the sheer part. The main typical example of a sheer module is a polycontractible module, that is, a finite direct product of modules, each of which is contracted by some group element. We show that every sheer module has a ‘large’ polycontractible submodule, in some suitable sense. We apply this to the study of compactly generated metabelian groups. For instance, we prove that they always have a maximal compact normal subgroup, and we extend the Bieri–Strebel characterization of compactly presentable metabelian groups from the discrete case to this more general setting.
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- © The Author(s), 2025. The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence
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