  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.
