On Topologically Big Divergent Trajectories

We study the behavior of $A$-orbits in $G/\Gamma$, when $G$ is a semisimple real algebraic $\mathbb{Q}$-group, $\Gamma$ is a non-uniform arithmetic lattice, and $A$ is a torus of dimension $\geq\operatorname{rank}_\mathbb{Q}(\Gamma)$. We show that every divergent trajectory of $A$ diverges due to a purely algebraic reason, % has a simple algebraic description. solving a longlasting conjecture of Weiss. In addition, we examine the intersections of $A$-orbits and show that in many cases every $A$-orbit intersects every deformation retract $X\subseteq G/\Gamma$. This solves the questions raised by Pettet and Souto. The proofs use algebraic and differential topology, as well as algebraic group theory.