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- Von_Neumann_conjecture abstract "In mathematics, the von Neumann conjecture stated that a group G is non-amenable if and only if G contains a subgroup that is a free group on two generators. The conjecture was disproved in 1980.In 1929, during his work on the Banach–Tarski paradox, John von Neumann defined the concept of amenable groups and showed that no amenable group contains a free subgroup of rank 2. The suggestion that the converse might hold, that is, that every non-amenable group contains a free subgroup on two generators, was made by a number of different authors in the 1950s and 1960s. Although von Neumann's name is popularly attached to the conjecture, its first written appearance seems to be due to Mahlon Day in 1957.The Tits alternative is a fundamental theorem which, in particular, establishes the conjecture within the class of linear groups.The general conjecture was shown to be false in 1980 by Alexander Ol'shanskii; he demonstrated that the Tarski monster group, which is easily seen not to have a free subgroup of rank 2, is not amenable. Two years later, Sergei Adian showed that certain Burnside groups are also counterexamples. None of these counterexamples are finitely presented, and for some years it was considered possible that the conjecture held for finitely presented groups. However, in 2003, Ol'shanskii and Mark Sapir exhibited a collection of finitely-presented groups which do not satisfy the conjecture.".
- Von_Neumann_conjecture wikiPageID "662256".
- Von_Neumann_conjecture wikiPageRevisionID "605830307".
- Von_Neumann_conjecture hasPhotoCollection Von_Neumann_conjecture.
- Von_Neumann_conjecture subject Category:Combinatorial_group_theory.
- Von_Neumann_conjecture subject Category:Disproved_conjectures.
- Von_Neumann_conjecture subject Category:Geometric_group_theory.
- Von_Neumann_conjecture subject Category:Topological_groups.
- Von_Neumann_conjecture type Abstraction100002137.
- Von_Neumann_conjecture type Cognition100023271.
- Von_Neumann_conjecture type Concept105835747.
- Von_Neumann_conjecture type Content105809192.
- Von_Neumann_conjecture type DisprovedConjectures.
- Von_Neumann_conjecture type Group100031264.
- Von_Neumann_conjecture type Hypothesis105888929.
- Von_Neumann_conjecture type Idea105833840.
- Von_Neumann_conjecture type PsychologicalFeature100023100.
- Von_Neumann_conjecture type Speculation105891783.
- Von_Neumann_conjecture type TopologicalGroups.
- Von_Neumann_conjecture comment "In mathematics, the von Neumann conjecture stated that a group G is non-amenable if and only if G contains a subgroup that is a free group on two generators. The conjecture was disproved in 1980.In 1929, during his work on the Banach–Tarski paradox, John von Neumann defined the concept of amenable groups and showed that no amenable group contains a free subgroup of rank 2.".
- Von_Neumann_conjecture label "Congettura di von Neumann".
- Von_Neumann_conjecture label "Conjetura de von Neumann".
- Von_Neumann_conjecture label "Von Neumann conjecture".
- Von_Neumann_conjecture sameAs Conjetura_de_von_Neumann.
- Von_Neumann_conjecture sameAs Congettura_di_von_Neumann.
- Von_Neumann_conjecture sameAs m.030r8h.
- Von_Neumann_conjecture sameAs Q939072.
- Von_Neumann_conjecture sameAs Q939072.
- Von_Neumann_conjecture sameAs Von_Neumann_conjecture.
- Von_Neumann_conjecture wasDerivedFrom Von_Neumann_conjecture?oldid=605830307.
- Von_Neumann_conjecture isPrimaryTopicOf Von_Neumann_conjecture.