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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