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 $\mathbb{Q}$, $\Gamma$ be an arithmetic subgroup of $G$. A trajectory in $G/\Gamma$ 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 $\mathbb{Q}$-factor of $G$ is of small enough dimension (relatively to the $\mathbb{Q}$-rank of the $\mathbb{Q}$-factor), then there are non-obvious divergent trajectories for the action of $A$ on $G/\Gamma$.