  We use a functional renormalization group approach to study the instabilities
due to electron-electron interactions in a bilayer honeycomb lattice model with
AA stacking, as it might be relevant for layered graphene with this structure.
Starting with a tight- binding description for the four $\pi$-bands, we
integrate out the modes of the dispersion by successively lowering an infrared
cutoff and determine the leading tendencies in the effective interactions. The
antiferromagnetic spin-density wave is an expected instability for dominant
local repulsion among the electrons, but for nonlocal interaction terms also
other instabilities occur. We discuss the phase diagrams depending on the model
parameters. We compare our results to single-layer graphene and the more common
AB-stacked bilayer, both qualitatively and quantitatively.
