On the product of two (totally) minimal topological groups and the three-space-problem

Eberhardt, Volker; Dierolf, Susanne; Schwanengel, Ulrich

doi:10.1007/bf01536179

Cited by 23 publications

(10 citation statements)

References 3 publications

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“…The next fact was originally proved in [12], and we use it in the subsequent Theorem 6.23. Fact 6.22.…”

Section: Non-torsionfree Case Given An Actionmentioning

confidence: 89%

“…A Hausdorff topological group (G, τ ) is called minimal if there exists no Hausdorff group topology on G which is strictly coarser than τ (see [11,31]). This class of groups, containing all compact ones, was largely studied in the last five decades, (see the papers [2,8,9,12,24,28], the surveys [4,32] and the book [7]). Since it is not stable under taking quotients, the following stronger notion was introduced in [6]: a minimal group G is totally minimal, if the quotient group G N is minimal for every closed normal subgroup N of G. This is precisely a group G satisfying the open mapping theorem, i.e., every continuous surjective homomorphism with domain G and codomain any Hausdorff topological group is open.…”

Section: November 9 2018 1 Introductionmentioning

confidence: 99%

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Hereditarily minimal topological groups

Dikranjan

Shlossberg

et al. 2019

Forum Mathematicum

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We study locally compact groups having all subgroups minimal. We call such groups hereditarily minimal. In 1972 Prodanov proved that the infinite hereditarily minimal compact abelian groups are precisely the groups Zp of p-adic integers. We extend Prodanov's theorem to the non-abelian case at several levels. For infinite hypercentral (in particular, nilpotent) locally compact groups we show that the hereditarily minimal ones remain the same as in the abelian case. On the other hand, we classify completely the locally compact solvable hereditarily minimal groups, showing that in particular they are always compact and metabelian.The proofs involve the (hereditarily) locally minimal groups, introduced similarly. In particular, we prove a conjecture by He, Xiao and the first two authors, showing that the group Qp ⋊ Q * p is hereditarily locally minimal, where Q * p is the multiplicative group of non-zero p-adic numbers acting on the first component by multiplication. Furthermore, it turns out that the locally compact solvable hereditarily minimal groups are closely related to this group.

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“…The next fact was originally proved in [12], and we use it in the subsequent Theorem 6.23. Fact 6.22.…”

Section: Non-torsionfree Case Given An Actionmentioning

confidence: 89%

Section: November 9 2018 1 Introductionmentioning

confidence: 99%

Hereditarily minimal topological groups

Dikranjan

Shlossberg

et al. 2019

Forum Mathematicum

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“…Fact D. ([7, (7)]) If a topological group G contains a compact normal subgroup N such that G/N is totally minimal, then G is totally minimal.…”

Section: Proofmentioning

confidence: 99%

Hereditarily h-complete groups

Lukács

2004

Topology and its Applications

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A topological group G is h-complete if every continuous homomorphic image of G is (Raikov-)complete; we say that G is hereditarily h-complete if every closed subgroup of G is h-complete. In this paper, we establish open-map properties of hereditarily h-complete groups with respect to large classes of groups, and prove a theorem on the (total) minimality of subdirectly represented groups. Numerous applications are presented, among them: 1. Every hereditarily h-complete group with quasi-invariant basis is the projective limit of its metrizable quotients; 2. If every countable discrete hereditarily h-complete group is finite, then every locally compact hereditarily h-complete group that has small invariant neighborhoods is compact. In the sequel, several open problems are formulated.Comment: 12 pages; few changes were made compared to the original submission thanks to the suggestions of the refere

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“…Indeed, there exist a minimal precompact group G and a twoelement subgroup P of Aut(G) such that G ⋋ P is not minimal. See Eberhardt-Dierolf-Schwanengel [11,Example 10] and also [8,Example 4.6]. The latter example also demonstrates that, in general, for two arbitrary minimal groups G and H the topological semidirect product G ⋋ H may fail to be minimal.…”

Section: Introductionmentioning

confidence: 97%

Minimality of the Semidirect Product

Megrelishvili,

Polev,

Shlossberg

2015

Preprint

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A topological group is minimal if it does not admit a strictly coarser Hausdorff group topology. We provide a sufficient and necessary condition for the minimality of the semidirect product G ⋋ P, where G is a compact topological group and P is a topological subgroup of Aut(G). We prove that G ⋋ P is minimal for every closed subgroup P of Aut(G). In case G is abelian, the same is true for every subgroup P ⊆ Aut(G). We show, in contrast, that there exist a compact two-step nilpotent group G and a subgroup P of Aut(G) such that G ⋋ P is not minimal. This answers a question of Dikranjan. Some of our results were inspired by a work of Gamarnik [12].

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On the product of two (totally) minimal topological groups and the three-space-problem

Cited by 23 publications

References 3 publications

Hereditarily minimal topological groups

Hereditarily minimal topological groups

Hereditarily h-complete groups

Minimality of the Semidirect Product

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