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 analogue 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. The first talk may be found here.
This video is part of the Institute for Advanced Study‘s Number theory seminar.
