Presentations of Galois groups of maximal extensions with restricted ramification

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 GS (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 BS (k, A), for each finite simple GS (k) -module A, to generalize the work of Koch about the pro- completion of GS (k) to study the whole group GS (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.