  We study the effect of electron-electron interactions in the electronic
properties of a biased graphene bilayer. This system is a semiconductor with
conduction and valence bands characterized by an unusual ``mexican-hat''
dispersion. We focus on the metallic regime where the chemical potential lies
in the ``mexican-hat'' in the conduction band, leading to a topologically
non-trivial Fermi surface in the shape of a ring. We show that due to the
unusual topology of the Fermi surface electron-electron interactions are
greatly enhanced. We discuss the possibility of an instability towards a
ferromagnetic phase due to this enhancement. We compute the electronic
polarization function in the random phase approximation and show that, while at
low energies the system behaves as a Fermi liquid (albeit with peculiar Friedel
oscillations), at high frequencies it shows a highly anomalous response when
compare to ordinary metals.
