Existence of non-obvious divergent trajectories in homogeneous spaces

Abstract

We prove a modified version for a conjecture of Weiss from 2004. Let G be a semisimple real algebraic group defined over ℚ, Γ be an arithmetic subgroup of G. A trajectory in G/Γ is divergent if eventually it leaves every compact subset, and is obvious divergent if there is a finite collection of algebraic data which cause the divergence. Let A be a diagonalizable subgroup of G of positive dimension. We show that if the projection of A to any ℚ-factor of G is of small enough dimension (relatively to the ℚ-rank of the ℚ-factor), then there are non-obvious divergent trajectories for the action of A on G/Γ.