C-minimal topological groups
-
Wenfei Xi
and
Menachem Shlossberg
Abstract
In this paper, we study topological groups having all closed subgroups (totally) minimal and we call such groups c-(totally) minimal. We show that a locally compact c-minimal connected group is compact. Using a well-known theorem of [P. Hall and C. R. Kulatilaka, A property of locally finite groups, J. Lond. Math. Soc. 39 1964, 235–239] and a characterization of a certain class of Lie groups, due to [S. K. Grosser and W. N. Herfort, Abelian subgroups of topological groups, Trans. Amer. Math. Soc. 283 1984, 1, 211–223], we prove that a c-minimal locally solvable Lie group is compact.
It is shown that a topological group G is c-(totally) minimal if and only if G has a compact normal subgroup N such that
Moreover, a c-totally minimal group that is either complete solvable or strongly compactly covered must be compact. Negatively answering [D. Dikranjan and M. Megrelishvili, Minimality conditions in topological groups, Recent Progress in General Topology. III, Atlantis Press, Paris 2014, 229–327, Question 3.10 (b)], we find, in contrast, a totally minimal solvable (even metabelian) Lie group that is not compact.
Funding source: National Natural Science Foundation of China
Award Identifier / Grant number: 12001264
Funding source: Natural Science Foundation of Jiangsu Province
Award Identifier / Grant number: BK20200834
Funding statement: The first author is supported by the National Natural Science Foundation of China (Grant No. 12001264) and the Natural Science Foundation of Jiangsu Province (Grant No. BK20200834).
Acknowledgements
It is a pleasure to thank the referee for a very careful reading and a wealth of helpful comments that improved this paper substantially. We thank M. Megrelishvili whose question motivated this research. Thanks are due also to D. Dikranjan, D. Peng and D. Toller for their useful suggestions.
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Articles in the same Issue
- Frontmatter
- On the topological complexity of manifolds with abelian fundamental group
- Reconstructing étale groupoids from semigroups
- Fiberwise linear differential operators
- Foliations with isolated singularities on Hirzebruch surfaces
- On a stronger reconstruction notion for monoids and clones
- Cochain level May–Steenrod operations
- Commutative L-algebras and measure theory
- Rankin–Selberg integrals for principal series representations of GL(n)
- A simple proof of the generalized Leibniz rule on bounded Euclidean domains
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- Seshadri constants on some Quot schemes
- Hörmander Fourier multiplier theorems with optimal Besov regularity on multi-parameter Hardy spaces
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- C-minimal topological groups
- Cevian properties in ideal lattices of Abelian ℓ-groups
- Study of twisted Bargmann transform via Bargmann transform
Articles in the same Issue
- Frontmatter
- On the topological complexity of manifolds with abelian fundamental group
- Reconstructing étale groupoids from semigroups
- Fiberwise linear differential operators
- Foliations with isolated singularities on Hirzebruch surfaces
- On a stronger reconstruction notion for monoids and clones
- Cochain level May–Steenrod operations
- Commutative L-algebras and measure theory
- Rankin–Selberg integrals for principal series representations of GL(n)
- A simple proof of the generalized Leibniz rule on bounded Euclidean domains
- Construction of a class of maximal commutative subalgebras of prime Leavitt path algebras
- Seshadri constants on some Quot schemes
- Hörmander Fourier multiplier theorems with optimal Besov regularity on multi-parameter Hardy spaces
- On spectral and non-spectral problem for the planar self-similar measures with four element digit sets
- C-minimal topological groups
- Cevian properties in ideal lattices of Abelian ℓ-groups
- Study of twisted Bargmann transform via Bargmann transform
