A nonhomogeneous orbit closure of a diagonal subgroup

Abstract

Let $G={\bf SL}l(n,\Bbb{R})$ with $n\geq 6$. We construct examples of lattices $\Gamma \subset G$, subgroups $A$ of the diagonal group $D$ and points $x\in G/\Gamma$ such that the closure of the orbit $Ax$ is not homogeneous and such that the action of $A$ does not factor through the action of a one-parameter nonunipotent group. This contradicts a conjecture of Margulis.