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 Comm_p(F), which is the commensurator of F viewed as a group with pro-p topology. By contrast with Comm(F), we prove that Comm_p(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 Comm_p(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 Comm_p(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.