Openly Haar null sets and conjugacy in Polish groups

Cohen, Michael P.; Kallman, Robert R.

doi:10.1007/s11856-016-1374-y

Cited by 6 publications

(16 citation statements)

References 15 publications

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“…Proof It is trivial that both

scriptHN

and

scriptGHN

satisfy (I) and (II) in Definition 3.2.1. The proof of (III) that is reproduced here is from the Appendix of [26], where a corrected version of the proof in [75] is given. Proving this fact is easier in abelian Polish groups (see [23, Theorem 1]) and when the group is metrizable with a complete left invariant metric (this would allow the proof of [75, Theorem 3] to work without modifications).…”

Section: Basic Propertiesmentioning

confidence: 99%

“…Proving this fact is easier in abelian Polish groups (see [23, Theorem 1]) and when the group is metrizable with a complete left invariant metric (this would allow the proof of [75, Theorem 3] to work without modifications). The Appendix of [26] mentions the other approaches and discusses the differences between them. The proof of (III) for Haar null and for generalized Haar null sets is very similar.…”

Section: Basic Propertiesmentioning

confidence: 99%

“…Proof It is trivial that the system of openly Haar null sets satisfies (I) and (II) in Definition 3.2.1. The following proof of (III) is described in the Appendix of [26] and is similar to the proof of Theorem 3.2.5. Let

A_{n}

be openly Haar null for all

n \in ω

, we prove that

A = ⋃_{n \in ω} A_{n}

is also openly Haar null.…”

Section: Alternative Definitionsmentioning

confidence: 99%

“…Unfortunately, there are groups where this ideal contains only the empty set. The following results about this ‘collapse’ are proved as [26, Propositons 5 and 3]: Proposition Assume that

G

is a Polish group. If for every compact subset

C \subseteq G

and every nonempty open subset

U \subseteq G

there are

g, h \in G

with

g C h \subseteq U

, then the empty set is the only openly Haar null subset of

G

.…”

Section: Alternative Definitionsmentioning

confidence: 99%

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Haar null and Haar meager sets: a survey and new results

Elekes

Nagy

2020

Bull. London Math. Soc.

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We survey results about Haar null subsets of (not necessarily locally compact) Polish groups. The aim of this paper is to collect the fundamental properties of the various possible definitions of Haar null sets, and also to review the techniques that may enable the reader to prove results in this area. We also present several recently introduced ideas, including the notion of Haar meager sets, which are closely analogous to Haar null sets. We prove some results in a more general setting than that of the papers where they were originally proved and prove some results for Haar meager sets which were already known for Haar null sets.

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“…Proof It is trivial that both

scriptHN

and

scriptGHN

Section: Basic Propertiesmentioning

confidence: 99%

Section: Basic Propertiesmentioning

confidence: 99%

A_{n}

be openly Haar null for all

n \in ω

, we prove that

A = ⋃_{n \in ω} A_{n}

is also openly Haar null.…”

Section: Alternative Definitionsmentioning

confidence: 99%

“…Unfortunately, there are groups where this ideal contains only the empty set. The following results about this ‘collapse’ are proved as [26, Propositons 5 and 3]: Proposition Assume that

G

is a Polish group. If for every compact subset

C \subseteq G

and every nonempty open subset

U \subseteq G

there are

g, h \in G

with

g C h \subseteq U

, then the empty set is the only openly Haar null subset of

G

.…”

Section: Alternative Definitionsmentioning

confidence: 99%

See 2 more Smart Citations

Haar null and Haar meager sets: a survey and new results

Elekes

Nagy

2020

Bull. London Math. Soc.

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“…It is known that the collection of Haar null sets forms a σ-ideal in every Polish group (see [4] and [16]) and it coincides with the ideal of measure zero sets in locally compact groups with respect to every left (or equivalently right) Haar measure. Using this definition, it makes sense to talk about the properties of random elements of a Polish group.…”

Section: Introductionmentioning

confidence: 99%

The structure of random automorphisms of countable structures

Darji

Elekes

Kalina

et al. 2019

Trans. Amer. Math. Soc.

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In order to understand the structure of the "typical" element of an automorphism group, one has to study how large the conjugacy classes of the group are. When typical is meant in the sense of Baire category, a complete description of the size of the conjugacy classes has been given by Kechris and Rosendal. Following Dougherty and Mycielski we investigate the measure theoretic dual of this problem, using Christensen's notion of Haar null sets. When typical means random, that is, almost every with respect to this notion of Haar null sets, the behavior of the automorphisms is entirely different from the Baire category case. In this paper, we generalize the theorems of Dougherty and Mycielski about S∞ to arbitrary automorphism groups of countable structures isolating a new model theoretic property, the Cofinal Strong Amalgamation Property. As an application we show that a large class of automorphism groups can be decomposed into the union of a meager and a Haar null set.2010 Mathematics Subject Classification. Primary 03E15, 22F50; Secondary 03C15, 28A05, 54H11, 28A99.

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