On commensurators of free groups and free pro-p groups

Abstract

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 $Com{m}_p(F)$, which is the commensurator of $F$ viewed as a group with pro-$p$ topology. By contrast with $Comm(F)$, we prove that $Com{m}_p(F)$ 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 $Com{m}_p(F)$ depends on the rank of $F$ and determine the exact dependency. If $\mathbf{F}$ is the pro-$p$ completion of $F$ (which is a free pro-$p$ group), $Comm(\mathbf{F})$ is a totally disconnected locally compact (tdlc) group containing $\mathbf{F}$ as an open subgroup. We use $Com{m}_p(F)$ to construct an abstractly simple subgroup of $Comm(\mathbf{F})$ containing $\mathbf{F}$ as well as a family of non-discrete tdlc groups which are compactly generated and simple.

58 pages. v3: minor changes based on the referee reports; to appear in Compositio Mathematica