  We use the functional renormalization group equation for the effective
average action to study the fixed point structure of gravity-fermion systems on
a curved background spacetime. We approximate the effective average action by
the Einstein-Hilbert action supplemented by a fermion kinetic term and a
coupling of the fermion bilinears to the spacetime curvature. The latter
interaction is singled out based on a "smart truncation building principle".
The resulting renormalization group flow possesses two families of interacting
renormalization group fixed points extending to any number of fermions. The
first family exhibits an upper bound on the number of fermions for which the
fixed points could provide a phenomenologically interesting high-energy
completion via the asymptotic safety mechanism. The second family comes without
such a bound. The inclusion of the non-minimal gravity-matter interaction is
crucial for discriminating the two families. Our work also clarifies the origin
of the strong regulator-dependence of the fixed point structure reported in
earlier literature and we comment on the relation of our findings to studies of
the same system based on a vertex expansion of the effective average action
around a flat background spacetime.
