Math 2.0

Abstract harmonic analysis Amenability and invariant means Conjecture

von Neumann conjecture

Builds on Amenable groupFree groupSubgroup

Statement

A group GG is not amenable if and only if GG contains a subgroup isomorphic to the free group on two generators.

Remarks

The direction "if" is true: a group that contains a free group of rank 22 is not amenable, as von Neumann showed when he introduced amenability. The direction "only if" is false. Ol'shanskii proved in 1980 that some Tarski monster groups are not amenable, and these groups have no free subgroups; see Ol'shanskii's counterexample. Adian showed in 1982 that free Burnside groups of large odd exponent are further counterexamples. Ol'shanskii and Sapir gave finitely presented counterexamples in 2003, and Monod gave counterexamples made of piecewise projective homeomorphisms of the line in 2013. The conjecture holds for finitely generated linear groups, by the Tits alternative. The name refers to von Neumann, but the first written statement appears to be Day's of 1957. Whether Thompson's group FF, which has no free subgroup of rank 22, is amenable is still open.

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Amenable group, concept, no evidence levelAmenable groupconceptFree group, concept, no evidence levelFree groupconceptSubgroup, concept, no evidence levelSubgroupconceptvon Neumann conjecture, conjecture, no evidence levelvon Neumann conjectureconjecture
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Amenable group concept
  2. Free group concept
  3. Subgroup concept
  4. von Neumann conjecture conjecture

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Further reading

  1. Web https://en.wikipedia.org/wiki/Von_Neumann_conjecture
About this page
Tag7K2N7P38TZ
URIm2://7K2N7P38TZ
Kindconjecture
WritingW0 stub
MSC43A07, 20F65, 20E05
StatusThis copy of the site has no build data. Run m2 status 7K2N7P38TZ.
AboutAmenable group
UsesAmenable group, Free group, Subgroup
Same as
Changed3 November 2026

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