  We investigate the instabilities of interacting electrons on the honeycomb
bilayer by means of the functional renormalization group for a range of
interactions up to the third-nearest neighbor. Besides a novel instability
toward a gapless charge-density wave we find that using interaction parameters
as determined by ab-initio calculations for graphene and graphite puts the
system close to the boundary between antiferromagnetic and quantum spin Hall
instabilities. Importantly, the energy scales for these instabilities are large
such that imperfections and deviations from the basic model are expected to
play a major role in real bilayer graphene, where interaction effects seem to
be seen only at smaller scales. We therefore analyze how reducing the critical
scale and small doping of the layers affect the instabilities.
