## Institute for Mathematical Sciences

## Preprint ims99-8

** E. Lindenstrauss and B. Weiss**
* On Sets Invariant under the Action of the Diagonal Group*

Abstract: We investigate closures of orbits for the action of the group of diagonal matrices acting on $SL(n,R)/SL(n,Z)$, where
$n \geq 3$. It has been conjectured by Margulis that possible
orbit-closures for this action are very restricted. Lending
support to this conjecture, we show that any orbit-closure
containing a compact orbit is homogeneous. Moreover if $n$ is
prime then any orbit whose closure contains a compact orbit is
either compact itself or dense. This implies a number-theoretic
result generalizing an isolation theorem of Cassels and
Swinnerton-Dyer for products of linear forms. We also obtain
similar results for other lattices instead of $SL(n,Z)$,
under a suitable irreducibility hypothesis.

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