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.