Von Neumann conjecture
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In mathematics, the von Neumann conjecture, disproved in recent years, 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.
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. Although his name is popularly attached to the conjecture, it does not seem that he believed the converse to be true. Rather, this suggestion was made by a number of different authors in the 1950s and 1960s; for example a statement attributed to M. Day from 1957.
The conjecture was shown to be false in 1980 by 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, Adjan 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 2000, Ol'shanskii and Sapir exhibited a collection of finitely-presented groups which do not satisfy the conjecture.