Von Neumann conjecture

In mathematics, the von Neumann conjecture stated that a topological group G is not amenable if and only if G contains a subgroup that is a free group on two generators. The conjecture was disproved in 1980.

In the 1920s, during his groundbreaking work on Banach spaces, John von Neumann showed that no amenable group contains a free subgroup of rank 2. The superficial similarity to the Tits alternative for matrix groups invited the suggestion that the converse (that every group that is not amenable contains a free subgroup on two generators) is true. Although von Neumann's name is popularly attached to the conjecture that the converse is true, it does not seem that von Neumann himself believed the converse to be true.^{[citation needed]} Rather, this suggestion was made by a number of different authors in the 1950s and 1960s, including in a statement attributed to Mahlon Day in 1957.

The 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.

References

• Adian, S. (1982), "Random walks on free periodic groups", Izv. Akad. Nauk SSSR, Ser. Mat. 46 (6): 1139–1149, 1343 . (Russian)
• Ol'shanskii, A. (1980), "On the question of the existence of an invariant mean on a group", Uspekhi Mat. Nauk 35 (4): 199–200 . (Russian)
• Ol'shanskii, A. & Sapir, M. (2003), "Non-amenable finitely presented torsion-by-cyclic groups", Publications Mathématiques de L'IHÉS 96 (1): 43–169, doi:10.1007/s10240-002-0006-7 .