On Discreteness of Commensurators

Abstract

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.

14 pages, 1 figure

References in corpus (2)

• Criteria for virtual fibering
• Commensurators of non-free finitely generated Kleinian groups

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