The Complexity of Topological Group Isomorphism

Kechris, Alexander S.; Nies, André; Tent, Katrin

doi:10.1017/jsl.2018.25

Cited by 13 publications

(38 citation statements)

References 12 publications

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“…By Lemma 2.4 (3,4) this is a Borel set. Now to finish the proof it suffices to intersect the latter with the family of compact groups accepting f , which is Borel by Lemma 2.8.…”

Section: Theorem 31 For Any Countable Sequencementioning

confidence: 85%

“…The following description of compact subsets/subgroups of S(ω) is a wellknown fact with an easy proof. It implies that the subset of F(S(ω)) consisting of all compact subgroups of S(ω) is Borel (see [3]). We will denote it by C. {Y i | i ∈ ω} be a partition of ω into pairwise disjoint nonempty finite subsets.…”

Section: Compact Subgroups Of Sf (ω) and Characteristicsmentioning

confidence: 99%

“…It is worth noting here that the structure of compact subgroups in S(ω) is quite complicated. For example it is proved in [3] that the isomorphism relation on the family of compact subgroups of S(ω) is as complicated as the isomorphisms relation on graphs. There subgroups are considered in the natural Borel space of subgroups of S(ω), see preliminaries below.…”

Section: Introductionmentioning

confidence: 99%

“…We mention here that the set U(S(ω)) of all closed subgroups of S(ω) is a Borel subset of F(S(ω)) (see Lemma 2.5 of [3]).…”

Section: Introductionmentioning

confidence: 99%

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Finitary shadows of compact subgroups of $$S(\omega )$$

Majcher-Iwanow

2020

Algebra Univers.

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Let LF be the lattice of all subgroups of the group SF (ω) of all finitary permutations of the set of natural numbers. We consider subgroups of SF (ω) of the form C ∩ SF (ω), where C is a compact subgroup of the group of all permutations. In particular, we study their distribution among elements of LF. We measure this using natural relations of orthogonality and almost containedness. We also study complexity of the corresponding families of compact subgroups of S(ω).

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“…By Lemma 2.4 (3,4) this is a Borel set. Now to finish the proof it suffices to intersect the latter with the family of compact groups accepting f , which is Borel by Lemma 2.8.…”

Section: Theorem 31 For Any Countable Sequencementioning

confidence: 85%

Section: Compact Subgroups Of Sf (ω) and Characteristicsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…We mention here that the set U(S(ω)) of all closed subgroups of S(ω) is a Borel subset of F(S(ω)) (see Lemma 2.5 of [3]).…”

Section: Introductionmentioning

confidence: 99%

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Finitary shadows of compact subgroups of $$S(\omega )$$

Majcher-Iwanow

2020

Algebra Univers.

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“…On questions affine to this topic see also the interesting recent work [5]. (2) We denote by K gf the class of graphs.…”

Section: Section 3 the Space Of Automorphism Groups Of A Borel Complmentioning

confidence: 99%

Some Results on Polish Groups

Paolini¹,

Shelah²

2020

Reports on Mathematical Logic

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We prove that no quantifier-free formula in the language of group theory can define the ℵ 1-half graph in a Polish group, thus generalising some results from [6]. We then pose some questions on the space of groups of automorphisms of a given Borel complete class, and observe that this space must contain at least one uncountable group. Finally, we prove some results on the structure of the group of automorphisms of a locally finite group: firstly, we prove that it is not the case that every group of automorphisms of a graph of power λ is the group of automorphism of a locally finite group of power λ; secondly, we conjecture that the group of automorphisms of a locally finite group of power λ has a locally finite subgroup of power λ, and reduce the problem to a problem on p-groups, thus settling the conjecture in the case λ = ℵ 0 .

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Network topology generation based on eigenvector centrality with real‐time guarantee

Wang

Xiao-yan

2022

Concurrency and Computation

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The real‐time performance is a major concern of complex real‐time systems during information acquisition, transmission, processing, and application. The interactions among information within these systems constitute complicated information exchange networks. Network topology plays an important role in the behaviors of information interactions and interferes, which severely affects the real‐time performance of the whole systems. Methods such as synchronization‐based method or traffic balancing strategy are the main ideas that guide the construction of network topology. However, these methods could not consider the real‐time performance in the process of topology construction. This article proposes an automatic topology generation algorithm with real‐time guarantee. Specifically speaking, the nodes in the network are clustered to generate the network topology by introducing the concepts of node degree and eigenvector centrality to ensure the real‐time performance of the network. The position of the nodes in the network is determined by comparing the node degree and the communication factors. Analytic method and simulation method are used to verify the real‐time performance of the proposed algorithm. Results show that the real‐time performance of more than half of the total information exchanges is improved compared with the traffic balancing method and topology grouping method for an industrial‐scale networking case.

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The Complexity of Topological Group Isomorphism

Cited by 13 publications

References 12 publications

Finitary shadows of compact subgroups of $$S(\omega )$$

Finitary shadows of compact subgroups of $$S(\omega )$$

Some Results on Polish Groups

Network topology generation based on eigenvector centrality with real‐time guarantee

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