University of Sydney Algebra Seminar

Colin Reid

Friday 12 September, 12-1pm, in Carslaw 175

Commensurators of free groups and free pro-p groups

(Joint work with Y. Barnea, M. Ershov, A. Le Boudec, M. Vannacci and Th. Weigel.) The commensurator of a group G is a group that encapsulates (up to a suitable equivalence) all isomorphisms between finite index subgroups of G. We study the commensurator of a free group F and of a free pro-p group, and also the p-commensurator of F (the subgroup of the commensurator that respects the pro-p topology on F), with a focus on normal subgroup structure; the p-commensurator turns out to have a simple subgroup of index at most 2. As well as 'global' results about the commensurator as a whole, we obtain some new constructions of simple groups: finitely generated simple groups with a free commensurated subgroup, and compactly generated simple locally compact groups that have an infinite pro-p open subgroup, possibly free pro-p. arXiv:2507.04120