Densely locally minimal groups

W. Xi; D. Dikranjan; M. Shlossberg; D. Toller

doi:10.1016/j.topol.2019.106846

Densely locally minimal groups

W. Xi, D. Dikranjan, M. Shlossberg, D. Toller

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Abstract

We study locally compact groups having all dense subgroups (locally) minimal. We call such groups densely (locally) minimal. In 1972 Prodanov proved that the infinite compact abelian groups having all subgroups minimal are precisely the groups Z_p of p-adic integers. In [30], we extended Prodanov's theorem to the non-abelian case at several levels. In this paper, we focus on the densely (locally) minimal abelian groups. We prove that in case that a topological abelian group G is either compact or connected locally compact, then G is densely locally minimal if and only if G either is a Lie group or has an open subgroup isomorphic to Z_p for some prime p. This should be compared with the main result of [9]. Our Theorem C provides another extension of Prodanov's theorem: an infinite locally compact group is densely minimal if and only if it is isomorphic to Z_p. In contrast, we show that there exists a densely minimal, compact, two-step nilpotent group that neither is a Lie group nor it has an open subgroup isomorphic to Z_p.

Original language	English
Article number	106846
Journal	Topology and its Applications
Volume	266
DOIs	https://doi.org/10.1016/j.topol.2019.106846
State	Published - 1 Oct 2019
Externally published	Yes

Bibliographical note

Publisher Copyright:
© 2019 Elsevier B.V.

Funding

The first-named author is supported by grant NSFC , number 11571175 , and takes this opportunity to thank Dikran Dikranjan for his generous hospitality and support. The second-named author is partially supported by grant PSD-2015-2017-DIMA - progetto PRID TokaDyMA of Udine University . The third and fourth-named authors are supported by Programma SIR 2014 by MIUR , project GADYGR, number RBSI14V2LI , cup G22I15000160008. The first-named author is supported by grant NSFC, number 11571175, and takes this opportunity to thank Dikran Dikranjan for his generous hospitality and support. The second-named author is partially supported by grant PSD-2015-2017-DIMA - progetto PRID TokaDyMA of Udine University. The third and fourth-named authors are supported by Programma SIR 2014 by MIUR, project GADYGR, number RBSI14V2LI, cup G22I15000160008.

Funders	Funder number
Udine University
National Natural Science Foundation of China	PSD-2015-2017-DIMA, 11571175
Ministero dell’Istruzione, dell’Università e della Ricerca	G22I15000160008
Università degli Studi di Udine

Keywords

Hereditarily (locally) minimal
Hilbert-Smith Conjecture
Lie group
Locally minimal group
Non-topologizable group
p-adic integer

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10.1016/j.topol.2019.106846

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title = "Densely locally minimal groups",

abstract = "We study locally compact groups having all dense subgroups (locally) minimal. We call such groups densely (locally) minimal. In 1972 Prodanov proved that the infinite compact abelian groups having all subgroups minimal are precisely the groups Zp of p-adic integers. In [30], we extended Prodanov's theorem to the non-abelian case at several levels. In this paper, we focus on the densely (locally) minimal abelian groups. We prove that in case that a topological abelian group G is either compact or connected locally compact, then G is densely locally minimal if and only if G either is a Lie group or has an open subgroup isomorphic to Zp for some prime p. This should be compared with the main result of [9]. Our Theorem C provides another extension of Prodanov's theorem: an infinite locally compact group is densely minimal if and only if it is isomorphic to Zp. In contrast, we show that there exists a densely minimal, compact, two-step nilpotent group that neither is a Lie group nor it has an open subgroup isomorphic to Zp.",

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note = "Publisher Copyright: {\textcopyright} 2019 Elsevier B.V.",

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AU - Toller, D.

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KW - Hilbert-Smith Conjecture

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KW - Locally minimal group

KW - Non-topologizable group

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Densely locally minimal groups

Abstract

Bibliographical note

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Keywords

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