A visit to 3-manifolds in the quest to understand random Galois groups

Cohen, Lenstra, and Martinet gave conjectural distributions for the class group of a random number field. Since the class group is the Galois group of the maximum abelian unramified extension, a natural generalization would be to give a conjecture for the distribution of the Galois group of the maximal unramified extension. Previous work (joint with Liu and Zurieck-Brown) produced a plausible conjecture for the part of this Galois group relatively prime to the number of roots of unity in the base field. There is a deep analogy between number fields and 3-manifolds. Thus, an analogous question would be to describe the distribution of the profinite completion of the fundamental group of a random 3-manifold. In this talk, I will explain how Will Sawin and I answered this question for a model of random 3-manifolds defined by Dunfield and Thurston, and how the techniques we used should allow us, in future work, to prove large q limit theorems in the function field analog and give a general conjecture in the number field case, taking into account roots of unity in the base field. This is part two of a series of two talks on joint work, some in progress, with Will Sawin. Both talks should be understandable on their own.