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On sets invariant under the action of the diagonal group

Published online by Cambridge University Press:  02 October 2001

ELON LINDENSTRAUSS
Affiliation:
Institute of Mathematics, Hebrew University, Jerusalem, Israel Present address: Department of Mathematics, Stanford University, Stanford, CA 94305, USA (e-mail: elon@member.ams.org)
BARAK WEISS
Affiliation:
Institute of Mathematical Sciences, SUNY, Stony Brook, NY 11794, USA Present address: Department of Mathematics, Ben Guryon University, Be'er Sheva, Israel 89105 (e-mail: barakw@cs.bgu.ac.il)

Abstract

We consider the action of the (n-1)-dimensional group of diagonal matrices in SL(n,\mathbb{R}) on SL(n,\mathbb{R})/\Gamma, where \Gamma is a lattice and n\ge 3. Far-reaching conjectures of Furstenberg, Katok–Spatzier and Margulis suggest that there are very few closed invariant sets for this action. We examine the closed invariant sets containing compact orbits. For example, for \Gamma={\rm SL}(n,\mathbb{Z}) we describe all possible orbit-closures containing a compact orbit. In marked contrast to the case n=2, such orbit-closures are necessarily homogeneous submanifolds in the sense of Ratner.

Type
Research Article
Copyright
2001 Cambridge University Press

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