High Quality Content by WIKIPEDIA articles! 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. 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.