Title: Presentations of Galois groups of maximal extensions with restricted ramifications

Abstract: In previous work with Melanie Matchett Wood and David Zureick-Brown, we conjecture that an explicitly-defined random profinite group model can predict the distribution of the Galois groups of maximal unramified extension of global fields that range over $\Gamma$-extensions of $\mathbb{Q}$ or $\mathbb{F}_q(t)$. In the function field case, our conjecture is supported by the moment computation, but very little is known in the number field case. Interestingly, our conjecture suggests that the random group should simulate the maximal unramified Galois groups, and hence suggests some particular requirements on the structure of these Galois groups. In this talk, we will prove that the maximal unramified Galois groups are always achievable by our random group model, which verifies those interesting requirements. The proof is motivated by the work of Lubotzky on the profinite presentations and by the work of Koch on the $p$-class tower groups. We will also discuss how the techniques used in the proof can be applied to the cases that are not covered by the Liu--Wood--Zureick-Brown conjecture, which potentially could help us obtain random group models for those cases.