question is then whether the quantum-gravitational model using “nonfoliated CDT" can reproduce the results of the standard formulation, especially those concerning the large-scale properties of quantum spacetime. The main conclusion of the present paper, previously announced in [@jl], is that it [*can*]{}, at least in space-time dimension 2+1. At least in higher dimensions, this result appears to weaken the potential link of CDT quantum gravity with Hořava-Lifshitz gravity [@hl; @phasediagram], where the presence of a preferred foliation is a key ingredient.[^2]

To summarize, our intention is to get rid of the [*distinguished*]{} foliation (and associated discrete time label $t$) of CDT, whose leaves for integer $t$ coincide with simplicial spatial hypermanifolds consisting entirely of [*space*]{}like subsimplices of codimension 1. This does not mean that the nonfoliated CDT configurations cannot in principle be foliated with respect to some continuous notion of time, but only that there is in general no canonical way of doing this in terms of a distinguished substructure of the triangulated spacetimes.

Of course, having [*a*]{} notion of time at the level of the regularized geometries is important. In the standard formulation of CDT, we get a notion of (discrete proper) time $t$ for free, simply by counting 