On commensurators of free groups and free pro-p groups

We study the commensurators of free groups and free pro-p groups, as well as certain subgroups of these. We prove that the commensurator Comm (F) of a non-abelian free group of finite rank F is not virtually simple, answering a question of Lubotzky. On the other hand, we exhibit a family of easy-to-define finitely generated subgroups of Comm (F) and show that some groups in this family are simple. For a prime p, we also consider the p-commensurator Commₚ (F), which is the commensurator of F viewed as a group with pro-p topology. By contrast with Comm (F), we prove that Commₚ (F) has a simple subgroup of index at most 2. Further, while the isomorphism class of Comm (F) does not depend on the rank of F, we prove that the isomorphism class of Commₚ (F) depends on the rank of F and determine the exact dependency. If F is the pro-p completion of F (which is a free pro-p group), Comm (F) is a totally disconnected locally compact (tdlc) group containing F as an open subgroup. We use Commₚ (F) to construct an abstractly simple subgroup of Comm (F) containing F as well as a family of non-discrete tdlc groups which are compactly generated and simple.