  The interpolated free group factors L(F_r), 1 < r <= \infty, are defined and
proofs of their properties with respect to compression by projections and
taking free products are proved. Hence it follows that all the free group
factor are isomorphic to each other or none of them are. These factors were
defined and these properties were proved independently by F. Radulescu, and
those given in this paper are equivalent, but use different techniques.
Specifically, we develop algebraic techniques that allow us to show that R*R =
L(F_2), where R is the hyperfinite II_1 factor.
