  We begin by showing that commensurators of Zariski dense subgroups of
isometry groups of symmetric spaces of non-compact 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 non-empty 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 $Isom ({\mathbb{H}}^3)$, commensurators are
discrete. Together these prove discreteness of commensurators for all known
examples of finitely generated, Zariski dense, infinite covolume discrete
subgroups of $Isom(X)$ for $X$ a symmetric space of non-compact type.
