scholarly journals Naively Haar null sets in Polish groups

2017 ◽  
Vol 446 (1) ◽  
pp. 193-200 ◽  
Author(s):  
Márton Elekes ◽  
Zoltán Vidnyánszky
Keyword(s):  
Polish Groups ◽  
Haar Null Sets

scholarly journals CONTINUITY OF UNIVERSALLY MEASURABLE HOMOMORPHISMS

2019 ◽  
Vol 7 ◽  
Author(s):  
CHRISTIAN ROSENDAL
Keyword(s):  
New York ◽  
Functional Analysis ◽  
Chromatic Number ◽  
Set Theory ◽  
Measure Theory ◽  
Polish Groups ◽  
Hamming Graph ◽  
Borel Structure ◽  
North Holland Publishing ◽  
Haar Null Sets

Answering a longstanding problem originating in Christensen’s seminal work on Haar null sets [Math. Scand. 28 (1971), 124–128; Israel J. Math. 13 (1972), 255–260; Topology and Borel Structure. Descriptive Topology and Set Theory with Applications to Functional Analysis and Measure Theory, North-Holland Mathematics Studies, 10 (Notas de Matematica, No. 51). (North-Holland Publishing Co., Amsterdam–London; American Elsevier Publishing Co., Inc., New York, 1974), iii+133 pp], we show that a universally measurable homomorphism between Polish groups is automatically continuous. Using our general analysis of continuity of group homomorphisms, this result is used to calibrate the strength of the existence of a discontinuous homomorphism between Polish groups. In particular, it is shown that, modulo $\text{ZF}+\text{DC}$ , the existence of a discontinuous homomorphism between Polish groups implies that the Hamming graph on $\{0,1\}^{\mathbb{N}}$ has finite chromatic number.

Openly Haar null sets and conjugacy in Polish groups

2016 ◽  
Vol 215 (1) ◽  
pp. 1-30 ◽  
Author(s):  
Michael P. Cohen ◽  
Robert R. Kallman
Keyword(s):  
Polish Groups ◽  
Haar Null Sets

scholarly journals AMENABILITY, FREE SUBGROUPS, AND HAAR NULL SETS IN NON-LOCALLY COMPACT GROUPS

2006 ◽  
Vol 93 (3) ◽  
pp. 693-722 ◽  
Author(s):  
SŁAWOMIR SOLECKI
Keyword(s):  
Locally Compact Groups ◽  
Compact Groups ◽  
Inverse Limits ◽  
Automatic Continuity ◽  
Locally Compact ◽  
Polish Groups ◽  
Continuity Result ◽  
Haar Null Sets ◽  
The One ◽  
Free Subgroups

The paper has two objectives. On the one hand, we study left Haar null sets, a measure-theoretic notion of smallness on Polish, not necessarily locally compact, groups. On the other hand, we introduce and investigate two classes of Polish groups which are closely related to this notion and to amenability. We show that left Haar null sets form a $\sigma$-ideal and have the Steinhaus property on Polish groups which are ‘amenable at the identity’, and that they lose these two properties in the presence of appropriately embedded free subgroups. As an application we prove an automatic continuity result for universally measurable homomorphisms from inverse limits of sequences of amenable, locally compact, second countable groups to second countable 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.

scholarly journals AUTOMORPHISM GROUPS OF RANDOMIZED STRUCTURES

2017 ◽  
Vol 82 (3) ◽  
pp. 1150-1179 ◽  
Author(s):  
TOMÁS IBARLUCÍA
Keyword(s):  
Automorphism Group ◽  
Automorphism Groups ◽  
Original Structure ◽  
Polish Groups ◽  
Separable Models

AbstractWe study automorphism groups of randomizations of separable structures, with focus on the ℵ0-categorical case. We give a description of the automorphism group of the Borel randomization in terms of the group of the original structure. In the ℵ0-categorical context, this provides a new source of Roelcke precompact Polish groups, and we describe the associated Roelcke compactifications. This allows us also to recover and generalize preservation results of stable and NIP formulas previously established in the literature, via a Banach-theoretic translation. Finally, we study and classify the separable models of the theory of beautiful pairs of randomizations, showing in particular that this theory is never ℵ0-categorical (except in basic cases).

scholarly journals Polish groups and Baire category methods

2016 ◽  
Vol 8 (1) ◽  
pp. 89-164 ◽  
Author(s):  
Julien Melleray
Keyword(s):  
Baire Category ◽  
Polish Groups

Ethnic identity of representatives of the Ukrainian minority university students in Poland

2021 ◽  
Vol 64 (21) ◽  
pp. 58-69
Author(s):  
Krzysztof Zegar ◽  
Maria Łoskot ◽  
Julia Pierzyńska ◽  
Małgorzata Siemiątkowska
Keyword(s):  
Decision Making ◽  
Ethnic Identity ◽  
National Identity ◽  
International Students ◽  
Social Identity ◽  
Identity Development ◽  
Narrative Analysis ◽  
Social Issues ◽  
Polish Groups ◽  
Self Confidence

Introduction: Referring to the knowledge about the number of Ukrainian students in Poland, James Marcia’s theory of identity development and Henri Tajfel’s theory of social identity, the authors examined how the Ukrainian minority studying in Poland describes its ethnic identity. Method: For this purpose, nine semistructural interviews were conducted, which were then subjected to a semantic narrative analysis. Results: It turned out that the respondents identify most strongly with the group of international students and students, and with their national identity in the second place. Polish nationality was cited as a group of belonging, spending time, while the Ukrainian nationality was individual, related to origin. Polish groups were positively evaluated by the respondents. The analysis also distinguished categories of differences between Poland and Ukraine, indicated by the respondents. They were: culture and religion, customs and tradition, decision-making and self-confidence, social issues, as well as mentality and science. The categories of stereotypes that were mentioned in the interviews were also identified: cheating and stealing, complaining and the similarity of nations. Conclusions: The results showed that the identity of Ukrainians is in a state of moratorium. The respondents define Ukraine as “their” country, while the strongest ones describe themselves as international students.

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