Biflatness and biprojectivity of the Fourier algebra

We show that the biflatness - in the sense of A. Ya. Helemskii - of the Fourier algebra $A(G)$ of a locally compact group $G$ forces $G$ to either have an abelian subgroup of finite index or to be non-amenable without containing $F_2$, the free group in two generators, as a closed subgroup. An analogous dichotomy is obtained for biprojectivity.