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We construct a novel Wetterich-type functional renormalization group equationfor gravity which encodes the gravitational degrees of freedom in terms ofgauge-invariant fluctuation fields. Applying a linear-geometric approximationthe structure of the new flow equation is considerably simpler than thestandard Quantum Einstein Gravity construction since only transverse-tracelessand trace part of the metric fluctuations propagate in loops. The geometricflow reproduces the phase-diagram of the Einstein-Hilbert truncation includingthe non-Gaussian fixed point essential for Asymptotic Safety. Extending theanalysis to the polynomial $f(R)$-approximation establishes that this fixedpoint comes with similar properties as the one found in metric Quantum EinsteinGravity; in particular it possesses three UV-relevant directions and is stablewith respect to deformations of the regulator functions by endomorphisms.

RG flows of Quantum Einstein Gravity in the linear-geometric approximation