Von Neumann conjecture
Encyclopedia
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 space
s, John von Neumann
showed that no amenable group
contains a free subgroup of rank 2. The superficial similarity to the Tits alternative
for matrix group
s 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.
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 counterexample
s. 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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the von Neumann conjecture stated that a topological group
Topological group
In mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the group's inverse function are continuous functions with respect to the topology. A topological group is a mathematical object with both an algebraic structure and a...
G is not amenable if and only if G contains a subgroup
Subgroup
In group theory, given a group G under a binary operation *, a subset H of G is called a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H x H is a group operation on H...
that is a free group
Free group
In mathematics, a group G is called free if there is a subset S of G such that any element of G can be written in one and only one way as a product of finitely many elements of S and their inverses...
on two generators. The conjecture was disproved in 1980.
In the 1920s, during his groundbreaking work on Banach space
Banach space
In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...
s, John von Neumann
John von Neumann
John von Neumann was a Hungarian-American mathematician and polymath who made major contributions to a vast number of fields, including set theory, functional analysis, quantum mechanics, ergodic theory, geometry, fluid dynamics, economics and game theory, computer science, numerical analysis,...
showed that no amenable group
Amenable group
In mathematics, an amenable group is a locally compact topological group G carrying a kind of averaging operation on bounded functions that is invariant under translation by group elements...
contains a free subgroup of rank 2. The superficial similarity to the Tits alternative
Tits alternative
In mathematics, the Tits alternative, named for Jacques Tits, is an important theorem about the structure of finitely generated linear groups. It states that every such group is either virtually solvable In mathematics, the Tits alternative, named for Jacques Tits, is an important theorem about...
for matrix group
Matrix group
In mathematics, a matrix group is a group G consisting of invertible matrices over some field K, usually fixed in advance, with operations of matrix multiplication and inversion. More generally, one can consider n × n matrices over a commutative ring R...
s 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.
The conjecture was shown to be false in 1980 by Alexander Ol'shanskii; he demonstrated that the Tarski monster group
Tarski monster group
In mathematics, a Tarski monster group, named for Alfred Tarski, is an infinite group G, such that every proper subgroup H of G, other than the identity subgroup, is a cyclic group of order a fixed prime number p. A Tarski monster group is necessarily simple. It was shown by A. Yu...
, which is easily seen not to have a free subgroup of rank 2, is not amenable. Two years later, Sergei Adian
Sergei Adian
Sergei Ivanovich Adian, also Adjan is one of the most prominent Soviet and Russian mathematicians. He is a professor at the Moscow State University. He is most famous for his work in group theory, especially on the Burnside problem.-Biography:...
showed that certain Burnside groups are also counterexample
Counterexample
In logic, and especially in its applications to mathematics and philosophy, a counterexample is an exception to a proposed general rule. For example, consider the proposition "all students are lazy"....
s. 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.