On sets invariant under the action of the diagonal group

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.