Geometry & Topology

On discreteness of commensurators

Mahan Mj

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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(3), commensurators are discrete. Together these prove discreteness of commensurators for all known examples of finitely presented, Zariski dense, infinite covolume discrete subgroups of Isom(X) for X an irreducible symmetric space of noncompact type.

Article information

Geom. Topol., Volume 15, Number 1 (2011), 331-350.

Received: 16 July 2010
Revised: 1 September 2010
Accepted: 15 December 2010
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M50: Geometric structures on low-dimensional manifolds

commensurator Cannon–Thurston map Kleinian group limit set


Mj, Mahan. On discreteness of commensurators. Geom. Topol. 15 (2011), no. 1, 331--350. doi:10.2140/gt.2011.15.331.

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