scholarly journals Minimal model-universal flows for locally compact Polish groups

2021 ◽  
Author(s):  
Colin Jahel ◽  
Andy Zucker
Keyword(s):  
Minimal Model ◽  
Locally Compact ◽  
Polish Groups

scholarly journals On Polish groups admitting non-essentially countable actions

2020 ◽  
pp. 1-15
Author(s):  
ALEXANDER S. KECHRIS ◽  
MACIEJ MALICKI ◽  
ARISTOTELIS PANAGIOTOPOULOS ◽  
JOSEPH ZIELINSKI
Keyword(s):  
Equivalence Relation ◽  
Isometry Group ◽  
Alternative Proof ◽  
Orbit Equivalence ◽  
Locally Compact ◽  
Continuous Action ◽  
Polish Groups ◽  
Infinite Dimensional ◽  
Open Question ◽  
New Criterion

Abstract It is a long-standing open question whether every Polish group that is not locally compact admits a Borel action on a standard Borel space whose associated orbit equivalence relation is not essentially countable. We answer this question positively for the class of all Polish groups that embed in the isometry group of a locally compact metric space. This class contains all non-archimedean Polish groups, for which we provide an alternative proof based on a new criterion for non-essential countability. Finally, we provide the following variant of a theorem of Solecki: every infinite-dimensional Banach space has a continuous action whose orbit equivalence relation is Borel but not essentially countable.

Actions of non-compact and non-locally compact Polish groups

2000 ◽  
Vol 65 (4) ◽  
pp. 1881-1894 ◽  
Author(s):  
Sławomir Solecki
Keyword(s):  
Equivalence Relation ◽  
Polish Space ◽  
Equivalence Relations ◽  
Orbit Equivalence ◽  
Locally Compact ◽  
Continuous Action ◽  
Borel Equivalence Relations ◽  
Polish Groups ◽  
Local Compactness ◽  
Polish Group

AbstractWe show that each non-compact Polish group admits a continuous action on a Polish space with non-smooth orbit equivalence relation. We actually construct a free such action. Thus for a Polish group compactness is equivalent to all continuous free actions of this group being smooth. This answers a question of Kechris. We also establish results relating local compactness of the group with its inability to induce orbit equivalence relations not reducible to countable Borel equivalence relations. Generalizing a result of Hjorth, we prove that each non-locally compact, that is, infinite dimensional, separable Banach space has a continuous action on a Polish space with non-Borel orbit equivalence relation, thus showing that this property characterizes non-local compactness among Banach spaces.

scholarly journals Ergodic multiplier properties

2014 ◽  
Vol 36 (3) ◽  
pp. 794-815 ◽  
Author(s):  
ADI GLÜCKSAM
Keyword(s):  
Irreducible Representation ◽  
Heisenberg Group ◽  
Weak Mixing ◽  
Locally Compact ◽  
Polish Groups ◽  
Infinite Dimensional ◽  
Dimensional Irreducible Representation ◽  
Weakly Mixing

In this article we will extend ‘the weak mixing theorem’ for certain locally compact Polish groups (Moore groups and minimally weakly mixing groups). In addition, we will show that the Gaussian action associated with the infinite-dimensional irreducible representation of the continuous Heisenberg group,$H_{3}(\mathbb{R})$, is weakly mixing but not mildly mixing.

scholarly journals UNIVERSAL SUBGROUPS OF POLISH GROUPS

2014 ◽  
Vol 79 (4) ◽  
pp. 1148-1183 ◽  
Author(s):  
KONSTANTINOS A. BEROS
Keyword(s):  
Topological Group ◽  
Banach Spaces ◽  
Topological Groups ◽  
Locally Compact ◽  
Polish Groups ◽  
Countable Power ◽  
Compact Case ◽  
Image Position ◽  
The Relationship ◽  
Compactly Generated

AbstractGiven a class${\cal C}$of subgroups of a topological groupG, we say that a subgroup$H \in {\cal C}$is auniversal${\cal C}$subgroupofGif every subgroup$K \in {\cal C}$is a continuous homomorphic preimage ofH. Such subgroups may be regarded as complete members of${\cal C}$with respect to a natural preorder on the set of subgroups ofG. We show that for any locally compact Polish groupG, the countable powerGωhas a universalKσsubgroup and a universal compactly generated subgroup. We prove a weaker version of this in the nonlocally compact case and provide an example showing that this result cannot readily be improved. Additionally, we show that many standard Banach spaces (viewed as additive topological groups) have universalKσand compactly generated subgroups. As an aside, we explore the relationship between the classes ofKσand compactly generated subgroups and give conditions under which the two coincide.

scholarly journals Equipment of Sets with Cardinality of the Continuum by Structures of Polish Groups with Haar Measures

2016 ◽  
Vol 5 ◽  
pp. 8-22
Author(s):  
Gogi Rauli Pantsulaia
Keyword(s):  
Haar Measure ◽  
Polish Space ◽  
Borel Probability Measure ◽  
Locally Compact ◽  
Polish Groups ◽  
Borel Measures ◽  
Isolated Points ◽  
Borel Probability ◽  
Group Structures ◽  
The Continuum

It is introduced a certain approach for equipment of sets with cardinality of the continuum by structures of Polish groups with two-sided (left or right) invariant Haar measures. By using this approach we answer positively Maleki’s certain question (2012) what are the real k-dimensional manifolds with at least two different Lie group structures that have the same Haar measure. It is demonstrated that for each diffused Borel probability measure defined in a Polish space (G;ρ;Bρ(G)) without isolated points there exist a metric ρ1and a group operation ⊙ in G such that Bρ(G) = Bρ1(G) and (G;ρ1;Bρ1(G);⊙) stands a compact Polish group with a two-sided (left or right) invariant Haar measure μ , where Bρ(G) and Bρ1(G) denote Borel σ-algebras of subsets of G generated by metrics ρ and ρ1, respectively. Similar approach is used for a construction of locally compact non-compact or non-locally compact Polish groups equipped with two-sided (left or right) invariant quasi-finite Borel measures.

scholarly journals Spatial models of Boolean actions and groups of isometries

2010 ◽  
Vol 31 (2) ◽  
pp. 405-421 ◽  
Author(s):  
ALEKSANDRA KWIATKOWSKA ◽  
SŁAWOMIR SOLECKI
Keyword(s):  
Spatial Model ◽  
Metric Spaces ◽  
Spatial Models ◽  
Locally Compact ◽  
Polish Groups ◽  
Recent Theorem ◽  
Groups Of Isometries ◽  
Hilbert’S Fifth Problem ◽  
Separable Metric Space

AbstractGiven a Polish group G of isometries of a locally compact separable metric space, we prove that each measure-preserving Boolean action by G has a spatial model or, in other words, has a point realization. This result extends both a classical theorem of Mackey and a recent theorem of Glasner and Weiss, and it covers interesting new examples. In order to prove our result, we give a characterization of Polish groups of isometries of locally compact separable metric spaces which may be of independent interest. The solution to Hilbert’s fifth problem plays an important role in establishing this characterization.

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