Volume 15, issue 1 (2011)

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On discreteness of commensurators

Mahan Mj

Geometry & Topology 15 (2011) 331–350

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.

commensurator, Cannon–Thurston map, Kleinian group, limit set
Mathematical Subject Classification 2000
Primary: 57M50
Received: 16 July 2010
Revised: 1 September 2010
Accepted: 15 December 2010
Published: 15 February 2011
Proposed: Benson Farb
Seconded: Walter Neumann, Jean-Pierre Otal
Mahan Mj
School of Mathematical Sciences
RKM Vivekananda University
PO Belur Math
Dt Howrah, WB-711202