On discreteness of commensurators

Abstract

We begin by showing that commensurators of Zariski dense subgroups of isometry groups of symmetric spaces of noncompact type are discrete provided that the limit set on the Furstenberg boundary is not invariant under the action of a (virtual) simple factor. In particular for rank one or simple Lie groups, Zariski dense subgroups with nonempty domain of discontinuity have discrete commensurators. This generalizes a Theorem of Greenberg for Kleinian groups. We then prove that for all finitely generated, Zariski dense, infinite covolume discrete subgroups of $Isom\left({ℍ}^3\right)$, commensurators are discrete. Together these prove discreteness of commensurators for all known examples of finitely presented, Zariski dense, infinite covolume discrete subgroups of $Isom\left(X\right)$ for $X$ an irreducible symmetric space of noncompact type.