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

Cohen and Lenstra gave a conjectural distribution for the  class group of a random quadratic 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 has produced a plausible conjecture in special cases,  with the most general being recent work of Liu, Wood, and Zurieck-Brown.

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 Melanie Wood 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 give a more general conjecture in the number field case.