The equivariant universality and couniversality of the Cantor cube

Megrelishvili, Michael; Scarr, Tzvi

doi:10.4064/fm167-3-4

Cited by 7 publications

(5 citation statements)

References 11 publications

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“…(2) It is well known that the symmetric group S N is universal for NA second countable groups. The same is true for the group of homeomorphisms Homeo(2 ω ), [24]. Therefore, S N is embedded into Homeo(2 ω ) and also Homeo(2 ω ) is embedded into S N .…”

Section: Corollary 54 the Homomorphismmentioning

confidence: 61%

“…For example, the Polish group S N is universal for all second countable (Polish) NA groups. The same is true for the Polish group H(2 ω ) (homeomorphism group of the Cantor cube 2 ω ), [24].…”

Section: Introductionmentioning

confidence: 76%

“…By results of [24], Homeo(2 ω ) is a universal Polish NA group. Moreover, the action of G := Homeo(2 ω ) on the Cantor cube 2 ω is universal in the class of all actions G × Y → Y , where Y is a metrizable Stone space and G is a topological subgroup of Homeo(Y ).…”

Section: Corollary 54 the Homomorphismmentioning

confidence: 95%

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Notes on non-archimedean topological groups

Megrelishvili

Shlossberg

2012

Topology and its Applications

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We study free topological groups defined over uniform spaces in some subclasses of the class NA of non-archimedean groups. Our descriptions of the corresponding topologies show that for metrizable uniformities the corresponding free balanced, free abelian and free Boolean NA groups are also metrizable. Graev type ultra-metrics determine the corresponding free topologies. Such results are in a striking contrast with free balanced and free abelian topological groups cases (in standard varieties).Another contrasting advantage is that the induced topological group actions on free abelian NA groups frequently remain continuous. One of the main applications is: any epimorphism in the category NA must be dense. Moreover, the same methods improve the following result of T.H. Fay [13]: the inclusion of a proper open subgroup H ֒→ G ∈ TGR is not an epimorphism in the category TGR of all Hausdorff topological groups. A key tool in the proofs is Pestov's test of epimorphisms [42].Our results provide a convenient way to produce surjectively universal NA abelian and balanced groups. In particular, we unify and strengthen some recent results of Gao [15] and Gao-Xuan [16] as well as classical results about profinite groups which go back to Iwasawa and Gildenhuys-Lim [17].

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Section: Corollary 54 the Homomorphismmentioning

confidence: 61%

Section: Introductionmentioning

confidence: 76%

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Notes on non-archimedean topological groups

Megrelishvili

Shlossberg

2012

Topology and its Applications

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“…(2) ⇒ (3) If the right uniformity µ is non-archimedean then by Lemma 3.1 the completion ( X, µ f in ) of its precompact replica (Samuel compactification of (X, µ)) is again non-archimedean. Now recall (see for example [35]) that this completion is just the greatest G-compactification β G G (the G-space analog of the Stone-Čech compactification) of G.…”

Section: 2mentioning

confidence: 99%

Notes on non-archimedean topological groups

Megrelishvili¹,

Shlossberg²

2010

Preprint

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We show that the Heisenberg type group H X = (Z 2 ⊕ V ) ⋋ V * , with the discrete Boolean group V := C(X, Z 2 ), canonically defined by any Stone space X, is always minimal. That is, H X does not admit any strictly coarser Hausdorff group topology. This leads us to the following result: for every (locally compact) non-archimedean G there exists a (resp., locally compact) nonarchimedean minimal group M such that G is a group retract of M. For discrete groups G the latter was proved by S. Dierolf and U. Schwanengel [8]. We unify some old and new characterization results for non-archimedean groups. We show also that any epimorphism into a non-archimedean group must be dense. Contents 1. Introduction and preliminaries 1 2. Minimality and group representations 3 3. Some facts about non-archimedean groups and uniformities 6 4. The Heisenberg group associated to a Stone space 9 5. More characterizations of non-archimedean groups 11 6. Automorphizable actions and epimorphisms in topological groups 12 References 20

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“…Thus (M, T ) is the unique universal minimal system (although not as a pointed system; fixing a distinguished point m 0 ∈ M and given a pointed minimal system (X, x 0 , T ), a homomorphism φ : L → X with φ(m 0 ) = x 0 may not exist). The next theorem is due to Pestov [6] (see also [5]). …”

Section: The Universal Transitive and Minimal Systems Of A Topologicamentioning

confidence: 99%

The universal minimal system for the group of homeomorphisms of the Cantor set

Glasner

Weiss

2003

Fund. Math.

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Abstract. Each topological group G admits a unique universal minimal dynamical system (M (G), G). For a locally compact noncompact group this is a nonmetrizable system with a rich structure, on which G acts effectively. However there are topological groups for which M (G) is the trivial one-point system (extremely amenable groups), as well as topological groups G for which M (G) is a metrizable space and for which one has an explicit description. We show that for the topological group G = Homeo(E) of self-homeomorphisms of the Cantor set E, with the topology of uniform convergence,

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The equivariant universality and couniversality of the Cantor cube

Abstract: Abstract. Let G, X, α be a G-space, where G is a non-Archimedean (having a local base at the identity consisting of open subgroups) and second countable topological group, and X is a zero-dimensional compact metrizable space. Let H({0,

Cited by 7 publications

References 11 publications

Notes on non-archimedean topological groups

Notes on non-archimedean topological groups

Notes on non-archimedean topological groups

The universal minimal system for the group of homeomorphisms of the Cantor set

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