Intersections of diagonal orbits

Abstract

Let $A\subseteq SL_n(\mathbb{R})$ group of diagonal matrices with positive diagonal, let ${\rm ST}_n\subseteq X_n:=SL_n(\mathbb{R})/SL_n(\mathbb{Z})$ be the set of stable lattices, and let ${\rm WR}_n\subseteq X_n$ be the set of well-rounded lattices. We prove that any $A$-orbit in $X_n$ intersects both ${\rm ST}_n$ and ${\rm WR}_n$.