In a fantastic paper from 2024, Yuan Liu, Melanie Matchett Wood and David Zureick-Brown constructed random group models to formulate conjectures on the distribution of Galois groups of maximal unramified extensions of global fields $K$, as $K$ varies over all $\Gamma$-extensions of $\mathbb{Q}$ or $\mathbb{F}_q(t)$. A consequence of these conjectures is that the Galois groups in question should satisfy certain interesting conditions. In a subsequent paper, Liu proved that Galois groups of maximal unramified extensions always admit a presentation in the form of this random group model, and that they satisfy the aforementioned requirements, providing evidence for the conjecture in Liu, Matchett Wood, Zureick-Brown. In this talk, I will sketch the proof of Liu’s beautiful result. If time permits, I will also discuss how Ravi and I used one of the objects defined in Liu’s paper to show that the dimension of the 2-Shafarevich group for a general module can increase if you increase the set.