Random pro-$p$ groups, braid groups, and random tame Galois groups

Abstract

We introduce a heuristic prediction for the distribution of the isomorphism class of the Galois group of the maximal pro-$p$ extension of $\mathbb{Q}$ unramified outside a “random” set of primes. This is guided by reasoning similar to that governing the Cohen–Lenstra conjectures. We conclude by describing theoretical and experimental evidence for our heuristic.