We study the behavior of A-orbits in G∕Γ when G is a semisimple real algebraic Q-group, Γ is a nonuniform arithmetic lattice, and A is a subgroup of the real torus of dimension ≥rankQG. We show that every divergent trajectory of A diverges due to a purely algebraic reason, solving a long-standing conjecture of Weiss. In addition, we examine sets that intersect all A-orbits, and show that in many cases every A-orbit intersects every deformation retract X⊂G∕Γ. This solves the questions raised by Pettet and Souto in 2009. The proofs use algebraic and differential topology, as well as algebraic group theory.