Geometric operators in the asymptotic safety scenario for quantum gravity

We consider geometric operators, such as the geodesic length and the volume of hypersurfaces, in the context of the asymptotic safety scenario for quantum gravity. We discuss the role of these operators from the asymptotic safety perspective, and compute their anomalous dimensions within the Einstein-Hilbert truncation. We also discuss certain subtleties arising in the definition of such geometric operators. Our results hint to an effective dimensional reduction of the considered geometric operators.