  We investigate the asymptotic safety conjecture for quantum gravity including
curvature invariants beyond Ricci scalars. Our strategy is put to work for
families of gravitational actions which depend on functions of the Ricci
scalar, the Ricci tensor, and products thereof. Combining functional
renormalisation with high order polynomial approximations and full numerical
integration we derive the renormalisation group flow for all couplings and
analyse their fixed points, scaling exponents, and the fixed point effective
action as a function of the background Ricci curvature. The theory is
characterised by three relevant couplings. Higher-dimensional couplings show
near-Gaussian scaling with increasing canonical mass dimension. We find that
Ricci tensor invariants stabilise the UV fixed point and lead to a rapid
convergence of polynomial approximations. We apply our results to models for
cosmology and establish that the gravitational fixed point admits inflationary
solutions. We also compare findings with those from $f(R)$-type theories in the
same approximation and pin-point the key new effects due to Ricci tensor
interactions. Implications for the asymptotic safety conjecture of gravity are
indicated.
