  In the paper, we study the generator problem of II$_1$ factors. By defining a
new concept related to the number of generators of a von Neumann algebra, we
are able to show that a large class of II$_1$ factors are singly generated,
i.e., generated by two self-adjoint elements. In particular, this shows that
most of II$_1$ factors, whose free entropy dimensions are less than or equal to
one, are singly generated.
