Existence of Non-Obvious Divergent Trajectories in homogeneous spaces

TL;DR

This paper proves the existence of non-obvious divergent trajectories in homogeneous spaces under certain algebraic conditions, extending Weiss's conjecture and revealing complex divergence behaviors in these dynamical systems.

Contribution

It establishes the existence of non-obvious divergent trajectories for diagonalizable group actions on homogeneous spaces under specific algebraic dimension constraints.

Findings

Existence of non-obvious divergent trajectories under certain conditions.

Extension of Weiss's conjecture to new algebraic settings.

Identification of algebraic dimension thresholds for divergence.

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 $G/\Gamma$.