Groups, Geometry, and Dynamics


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Volume 5, Issue 2, 2011, pp. 265–280
DOI: 10.4171/GGD/127

Published online: 2011-03-06

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

Nigel Boston[1] and Jordan S. Ellenberg[2]

(1) University of Wisconsin, Madison, United States
(2) University of Wisconsin, Madison, USA

We introduce a heuristic prediction for the distribution of the isomorphism class of the Galois group of the maximal pro-$p$ extension of $\mathbb{Q}$ unramified outside a “random” set of primes. This is guided by reasoning similar to that governing the Cohen–Lenstra conjectures. We conclude by describing theoretical and experimental evidence for our heuristic.

Keywords: Braid groups, random matrices, Cohen–Lenstra, Galois groups, restricted ramification, pro-$p$ groups

Boston Nigel, Ellenberg Jordan: Random pro-$p$ groups, braid groups, and random tame Galois groups. Groups Geom. Dyn. 5 (2011), 265-280. doi: 10.4171/GGD/127