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 $G_S(k)$, the Galois group 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 define a group $B_S(k,A)$, for each finite simple $G_S(k)$-module $A$, to generalize the work of Koch and Shafarevich on the pro-$\ell$ completion of $G_S(k)$. We prove that $G_S(k)$ always admits a balanced presentation when it is finitely generated. In the setting of the nonabelian Cohen–Lenstra heuristics, we prove that the unramified Galois groups studied by the Liu–Wood–Zureick-Brown conjecture always admit a balanced presentation in the form of the random group in the conjecture.