Motivated by the work of Lubotzky, we use Galois cohomology to study the
difference between the number of generators and the minimal number of relations
in a presentation of the Galois group GSβ(k) of the maximal extension of a
global field k that is unramified outside a finite set S of places, as k
varies among a certain family of extensions of a fixed global field Q. We
prove a generalized version of the global Euler-Poincar\'{e} Characteristic,
and define a group BSβ(k,A), for each finite simple GSβ(k)-module A, to
generalize the work of Koch about the pro-β completion of GSβ(k) to
study the whole group GSβ(k). In the setting of the nonabelian Cohen-Lenstra
heuristics, we prove that the objects studied by the Liu--Wood--Zureick-Brown
conjecture are always achievable by the random group that is constructed in the
definition the probability measure in the conjecture.Comment: Comments are welcom