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

In this article we introduce a heuristic prediction for the distribution of the isomorphism class of GS(p), the Galois group of the maximal pro-p extension of Q unramified outside of S, where S is a ”random” set of primes. That these groups should exhibit any statistical regularity is not at all obvious; our expectations in this direction are guided by the Cohen-Lenstra conjectures, which (among other things) predict quite precisely how often a fixed finite group appears as the ideal class group of a quadratic imaginary field. The Cohen-Lenstra conjectures can be obtained in (at least) two ways. On the one hand, the distribution on finite abelian groups suggested by the heuristics has a good claim on being the most natural “uniform distribution” on the category of finite abelian p-groups. On the other hand, as observed by Friedman and Washington ([8], see also [1]) the conjectures can also be recovered via the analogy between number fields and function fields; here one thinks of the class group as the cokernel of γ − 1 where γ is a p-adic matrix drawn randomly from a suitable subset of the Qp-points of an algebraic group. We will show that both heuristic arguments can be generalized to the nonabelian pro-p case, and that both lead to the same prediction, Heuristic 2.4 below. We conclude by describing some evidence, both theoretical and experimental, that supports (or at least is consistent with) Heuristic 2.4. We pay special attention to the interesting case where p = 2 and S consists of two primes congruent to 5 (mod 8). In this case, the heuristic appears to suggest that GS(p) is infinite 1/16 of the time.