A predicted distribution for Galois groups of maximal unramified extensions

We consider the distribution of the Galois groups Gal(K^un/K) of maximal unramified extensions as K ranges over Γ-extensions of ℚ or F_q(t). We prove two properties of Gal(K^un/K) coming from number theory, which we use as motivation to build a probability distribution on profinite groups with these properties. In Part I, we build such a distribution as a limit of distributions on n-generated profinite groups. In Part II, we prove as q→∞, agreement of Gal(K^un/K) as K varies over totally real Γ-extensions of F_q(t) with our distribution from Part I, in the moments that are relatively prime to q(q−1)|Γ|. In particular, we prove for every finite group Γ, in the q→∞ limit, the prime-to-q(q−1)|Γ|-moments of the distribution of class groups of totally real Γ-extensions of F_q(t) agree with the prediction of the Cohen–Lenstra–Martinet heuristics.