No abstract

Let G be a real algebraic group defined over Q, let be an arithmetic subgroup, and let T be any torus containing a maximal R-split torus. We prove that the closed orbits for the action of T on G/ admit a simple algebraic description. In particular, we show that if G is reductive, an orbit T x is closed if and only if x −1 T x is a product of a compact torus and a torus defined over Q, and it is divergent if and only if the maximal R-split subtorus of x −1 T x is defined over Q and Q-split. Our analysis also yields the following:• there is a compact K ⊂ G/ which intersects every T -orbit;• if rank Q G < rank R G, there are no divergent orbits for T .

We prove an S-arithmetic version, in the context of algebraic groups defined over number fields, of Ratner's theorem for closures of orbits of subgroups generated by unipotent elements. We apply this result in order to obtain a generalization of results of Margulis and of Borel-Prasad about values of irrational quadratic forms at integral points to the general setting of hermitian forms over division algebras with involutions of first or second kind. As a byproduct of our considerations we obtain another proof of the strong approximation theorem for algebraic groups defined over number fields.

No abstract

No abstract