Biflatness and biprojectivity of the Fourier algebra

Abstract.

We show that the biflatness—in the sense of A. Ya. Helemskiĭ—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 ${\mathbb{F}}_{2}$ as a closed subgroup. An analogous dichotomy is obtained for biprojectivity.