Type: Article
Amenability of group actions characterized by configuration equations
Journal: Bulletin of the Belgian Mathematical Society - Simon Stevin (13701444)Year: 2024Volume: Issue: 4Pages: 434 - 445
DOI:10.36045/j.bbms.231115aLanguage: English
Abstract
Extending a result by Rosenblatt and Willis, we present a new characterization for amenability of group actions. We show that a group action is amenable if and only if certain linear equations admit normalized solutions. We also give a new statement for the ping-pong Lemma to ensure that the group action is nonamenable and then obtain an upper bound for its Tarski number. © 2024 Belgian Mathematical Society. All rights reserved.
Author Keywords
AmenabilityConfigurationgroup actionParadoxical decompositionsTarski numberAbelian-by-finite groupsChermak-DelgadoCompact groupsComplex group algebrasHilbert space ℓ2 (G)Torsion-free groupsZero divisor conjectureDefinabilityelementary extensionslogical generationlogically cyclic groupsCompact groupNilpotent probabilityCommuting probabilityEngellocally (soluble-by-finite)Prüfer rankFours groupGroup algebrasTorsion-free groupUnit conjectureAutomorphismCoclassCoclass treep-group2 -closureNilpotent grouppermutation grouptotally 2 -closedGeneralized exponentpolynomial identitynoninner automorphismCamina pairfinite p-groupOuter automorphismP-automorphismP-groupsComplete multipartite graphIntegral connected Cayley graphNon-simple groupSimple groupCenter of a groupFinite p-groupsFrattini subgroupEngel set4-Engel groupsRight 4-Engel elements of a groupAbelian quotientAmenable groupConfiguration equivalent groupsConfiguration for a finitely generated groupCompletely reducible groupsCovering groups by subgroupsSimple groupsEngel groupsLeft Engel elementsNilpotent groupsFinite groupNoncyclic graphPermutable groupsRewritable groupsEngel conditionsFinite subsetsAbelian-by-finite groupsChermak-DelgadoCompact groupsComplex group algebrasHilbert space ℓ2 (G)Torsion-free groupsZero divisor conjectureDefinabilityelementary extensionslogical generationlogically cyclic groupsCompact groupNilpotent probabilityCommuting probabilityEngellocally (soluble-by-finite)Prüfer rankFours groupGroup algebrasTorsion-free groupUnit conjectureAutomorphismCoclassCoclass treep-group2 -closureNilpotent grouppermutation grouptotally 2 -closedGeneralized exponentpolynomial identitynoninner automorphismCamina pairfinite p-groupOuter automorphismP-automorphismP-groupsComplete multipartite graphIntegral connected Cayley graphNon-simple groupSimple groupCenter of a groupFinite p-groupsFrattini subgroupEngel set4-Engel groupsRight 4-Engel elements of a groupAbelian quotientAmenable groupConfiguration equivalent groupsConfiguration for a finitely generated groupCompletely reducible groupsCovering groups by subgroupsSimple groupsEngel groupsLeft Engel elementsNilpotent groupsFinite groupNoncyclic graphPermutable groupsRewritable groupsEngel conditionsFinite subsets