On Topologically Big Divergent Trajectories

Abstract

We study the behavior of A-orbits in G/Γ, when G is a semisimple real algebraic Q-group, Γ is a non-uniform arithmetic lattice, and A is a torus of dimension ≥ rank Q (Γ). We show that every divergent trajectory of A diverges due to a purely algebraic reason, solving a longlasting conjecture of Weiss [30, Conjecture 4.11]. In addition, we examine the intersections of 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 [21]. The proofs use algebraic and differential topology, as well as algebraic group theory.