  This is an expanded and revised version of our geometrical analysis of the
strong coupling phase of 4D simplicial quantum gravity. The main differences
with respect to the former version is a full discussion of singular
triangulations with singular vertices connected by a subsingular edge. In
particular we provide analytical arguments which characterize the entropical
properties of triangulations with a singular edge connecting two singular
vertices. The analytical estimate of the location of the critical coupling at
k_2\simeq 1.3093 is presented in more details. Finally we also provide a model
for pseudo-criticality at finite N_4(S^4).
