On Discreteness of Commensurators

TL;DR

This paper proves that under certain conditions, the commensurators of Zariski dense subgroups in isometry groups of symmetric spaces are discrete, extending previous results and covering all known cases of such groups.

Contribution

It generalizes Greenberg's theorem to higher rank symmetric spaces and establishes discreteness of commensurators for all known finitely generated, Zariski dense, infinite covolume discrete subgroups.

Findings

Commensurators are discrete if the limit set is not invariant under a simple factor.

For rank one or simple Lie groups, Zariski dense subgroups with non-empty domain of discontinuity have discrete commensurators.

Discreteness of commensurators holds for all known finitely generated, Zariski dense, infinite covolume discrete subgroups of isometry groups.

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.