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.