  In any dimension $D$, the Euclidean Einstein-Hilbert action, which describes
gravity in the absence of matter, can be discretized over random discrete
spaces obtained by gluing families of polytopes together in all possible ways.
In the physical limit of small Newton constant, only the spaces which maximize
the mean curvature survive. In two dimensions, this results in a theory of
random discrete spheres, which converge in the continuum limit towards the
Brownian sphere, a random fractal space interpreted as a quantum random
space-time. In this limit, the continuous Liouville theory of $D=2$ quantum
gravity is recovered.
  Previous results in higher dimension regarded triangulations - gluings of
tetrahedra or $D$-dimensional generalizations, leading to the continuum random
tree, or gluings of simple colored building blocks of small sizes, for which
multi-trace matrix model results are recovered. This work aims at providing
combinatorial tools which would allow a systematic study of richer building
blocks and of the spaces they generate in the continuum. We develop a bijection
with stacked two-dimensional discrete surfaces, and detail how it can be used
to classify discrete spaces according to their mean curvature and topology.
  A number of blocks are analyzed, including the new infinite family of
bi-pyramids, as well as toroidal and $D$-dimensional generalizations. The
relation to random tensor models is detailed. A central concern is the lowest
bound on the number of ($D-2$)-cells for any given blocks, or equivalently the
right scaling for the associated tensor model to have a well-behaved $1/N$
expansion. We also apply our bijection to the identification of the graphs
contributing at any order to the $2n$-point functions of the colored SYK model,
and to the enumeration of generalized unicellular maps - spaces obtained from a
single building block - according to their mean curvature.
