  Asymptotic safety is a promising mechanism for obtaining a consistent and
predictive quantum theory for gravity. The ADM formalism allows to introduce a
(Euclidean) time-direction in this framework. It equips spacetime with a
foliation structure by encoding the gravitational degrees of freedom in a lapse
function, shift vector, and a metric measuring distances on the spatial slices.
We use the Wetterich equation to study the renormalization group flow of the
graviton 2-point function extracted from the spatial metric. The flow is driven
by the 3- and 4-point vertices generated by the foliated Einstein-Hilbert
action supplemented by minimally coupled scalar and vector fields. We derive
bounds on the number of matter fields cast by asymptotic safety. Moreover, we
show that the phase diagram obtained in the pure gravity case is qualitatively
stable within these bounds. An intriguing feature is the presence of an
IR-fixed point for the graviton mass which prevents the squared mass taking
negative values. This feature persists for any number of matter fields and, in
particular, also in situations where there is no suitable interacting fixed
point rendering the theory asymptotically safe. Our work complements earlier
studies of the subject by taking contributions from the matter fields into
account.
