  We study the non-perturbative renormalisation of quantum gravity in four
dimensions. Taking care to disentangle physical degrees of freedom, we observe
the topological nature of conformal fluctuations arising from the functional
measure. The resulting beta functions possess an asymptotically safe fixed
point with a global phase structure leading to classical general relativity for
positive, negative or vanishing cosmological constant. If only the conformal
fluctuations are quantised we find an asymptotically safe fixed point
predicting a vanishing cosmological constant on all scales. At this fixed point
we reproduce the critical exponent, $\nu = 1/3$, found in numerical lattice
studies by Hamber. This suggests the fixed point may be physical while solving
the cosmological constant problem.
