No CrossRef data available.
Article contents
Locally compact modules over abelian groups and compactly generated metabelian groups
Part of:
Locally compact groups and their algebras
Chain conditions, finiteness conditions
Locally compact abelian groups
Theory of modules and ideals
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.
Information
- Type
- Research Article
- Information
- Copyright
- © The Author(s), 2025. The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence
References
Abels, H., Kompakt definierbare topologische Gruppen, Math. Ann. 197 (1972), 221–233.10.1007/BF01428228CrossRefGoogle Scholar
Baumgartner, U. and Willis, G., Contraction groups and scales of automorphisms of totally disconnected locally compact groups, Israel J. Math. 142 (2004), 221–248.10.1007/BF02771534CrossRefGoogle Scholar
Bieri, R. and Groves, J., The geometry of the set of characters induced by valuations, J. Reine Angew. Math. 347 (1984), 168–195.Google Scholar
Bieri, R., Neumann, W. and Strebel, R., A geometric invariant of discrete groups, Invent. Math. 90 (1987), 451–477.10.1007/BF01389175CrossRefGoogle Scholar
Bieri, R. and Strebel, R., Almost finitely presented soluble groups, Comment. Math. Helv. 53 (1978), 258–278.10.1007/BF02566077CrossRefGoogle Scholar
Bieri, R. and Strebel, R., Valuations and finitely presented metabelian groups, Proc. Lond. Math. Soc. (3) 41 (1980), 439–464.10.1112/plms/s3-41.3.439CrossRefGoogle Scholar
Conze, J.-P. and Guivarc’h, Y., Remarques sur la distalité dans les espaces vectoriels, C. R. Acad. Sci. Paris Ser. A 278 (1974), 1083–1086.Google Scholar
Cornulier, Y., Commability and focal locally compact groups, Indiana Univ. Math. J. 64 (2015), 115–150.10.1512/iumj.2015.64.5441CrossRefGoogle Scholar
Corson, H. and Glicksberg, I., Compactness in
$\operatorname{Hom}(G,H)$
, Canad. J. Math. 22 (1970), 164–170.10.4153/CJM-1970-019-1CrossRefGoogle Scholar
$\operatorname{Hom}(G,H)$
, Canad. J. Math. 22 (1970), 164–170.10.4153/CJM-1970-019-1CrossRefGoogle ScholarCornulier, Y. and de la Harpe, P., Metric geometry of locally compact groups, Tracts in Mathematics, vol. 25 (European Mathematical Society, 2016).Google Scholar
Glockner, H. and Willis, G., Classification of the simple factors appearing in composition series of totally disconnected contraction groups, J. Reine Angew. Math. 643 (2010), 141–169.Google Scholar
Siebert, E., Contractive automorphisms on locally compact groups, Math. Z. 191 (1986), 73–90.10.1007/BF01163611CrossRefGoogle Scholar
Willis, G., The structure of totally disconnected, locally compact groups, Math. Ann. 300 (1994), 341–363.10.1007/BF01450491CrossRefGoogle Scholar
Wu, T. S. and Yu, Y. K., Compactness properties of topological groups, Michigan Math. J. 19 (1972), 299–313.Google Scholar