shows part of a 2+1 dimensional CDT spacetime. Each leaf at integer-$t$ of the foliation forms a two-dimensional triangulation of the same fixed topology ${}^{(2)}\Sigma$, and consists of equilateral spacelike triangles with link length $\ell_s$. Adjacent triangulated spatial hypermanifolds are connected using Lorentzian tetrahedra to form a 2+1-dimensional simplicial manifold with spacetime topology $[0,1]\times {}^{(2)}\Sigma$, which again is a sequence of $t_{tot}$ triangulated “slabs" $\Delta t=1$. Each such geometry comes with a time label $t$ which can be defined continuously throughout the triangulation, and for integer values coincides with the discrete labeling of the simplicial leaves just described [@dl]. For reasons of simplicity, in the simulations time is often compactified. The topology then becomes $S^1\times {}^{(2)}\Sigma$.

The fundamental building blocks of three-dimensional CDT quantum gravity are flat Minkowskian tetrahedra, whose geometric properties are determined by their edge lengths. As in 1+1 dimensions, one considers two different link lengths, one spacelike with squared length $\ell_s^2$ and one timelike with $\ell_t^2=-\alpha \ell_s^2$. Without loss of generality we have introduced here a positive constant $\alpha$, which quantifies the relative magnitude of space- and timelike edge lengths and in what follows will be referred to as the *asymmetry parameter*.

CDT path integral configurations are 