  We discuss the renormalization of Einstein-Hilbert's gravity in
$d=2+\epsilon$ dimensions. We show that the application of the path-integral
approach leads naturally to scheme- and gauge-independent results on-shell, but
also gives a natural notion of quantum metric off-shell, which is the natural
argument of the effective action, even at the leading order in perturbation
theory. The renormalization group of Newton's constant is consistent with the
asymptotic safety scenario for quantum gravity in that it has an ultraviolet
relevant fixed point. We extend the approach to the analysis of curvature
square operators, understood as composites operators, which allows for the
determination of the spectrum of scaling operators at the scale invariant fixed
point. The analysis suggests that there is one operator that becomes relevant
close to $d=4$ dimensions, while other operators previously found in the
literature are either marginal or trivial on-shell.
