On discreteness of commensurators

Mj, Mahan

doi:10.2140/gt.2011.15.331

Download this article
	For screen
	For printing

Recent Issues

The Journal

Submission Guidelines

Submission Page

Policies for Authors

Ethics Statement

ISSN 1364-0380 (online)

ISSN 1465-3060 (print)

Author Index

To Appear

Other MSP Journals

On discreteness of commensurators

Mahan Mj

Geometry & Topology 15 (2011) 331–350

DOI: 10.2140/gt.2011.15.331

Abstract

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 $Isom (ℍ^{3})$ , 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.

Keywords

commensurator, Cannon–Thurston map, Kleinian group, limit set

Mathematical Subject Classification 2000

Primary: 57M50

References

Publication

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

Authors

Mahan Mj
School of Mathematical Sciences RKM Vivekananda University PO Belur Math Dt Howrah, WB-711202 India
http://people.rkmvu.ac.in/~mahan/

Volume 15, issue 1 (2011)