An isomorphism of the free group factors

We solve the free group factor isomorphism problem affirmatively by proving that $L(\F_2)$ and $L(\F_3)$ are isomorphic as tracial von Neumann algebras. The classical free group factor alternative then implies that all interpolated free group factors, including $L(\F_\infty)$, are isomorphic and have fundamental group $\R_{>0}$.