# Von Neumann Conjecture

## Conjecture

Let $G$ be a group.

Then $G$ is non-amenable if and only if $G$ contains a free subgroup of rank $2$.

## Resolution

This conjecture was answered in the negative, by Aleksandr Yuryevich Olshansky.

He demonstrated that the Tarski monster is a non-amenable group with no such free subgroup.

## Source of Name

This entry was named for John von Neumann.