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