Divergent Trajectories in Arithmetic Homogeneous Spaces of Rational Rank Two

TL;DR

This paper investigates the behavior of divergent trajectories in arithmetic homogeneous spaces of rational rank two, providing conditions for non-obvious divergence and a complete classification in this specific rank setting.

Contribution

It offers a sufficient condition for non-obvious divergence trajectories and fully classifies such trajectories when the rational and real ranks are both two.

Findings

Identifies when non-obvious divergence occurs in these spaces.

Provides a complete classification for rank-two cases.

Establishes a criterion linking algebraic data to divergence behavior.

Abstract

Let $G$ be a real algebraic group defined over $\mathbb{Q}$, $\Gamma$ be an arithmetic subgroup of $G$, and $T$ be a maximal $\mathbb{R}$-split torus. A trajectory in $G/\Gamma$ is divergent if eventually it leaves every compact subset. In some cases there is a finite collection of explicit algebraic data which account for the divergence. If this is the case, the divergent trajectory is called obvious. Given a closed cone in $T$, we study the existence of non-obvious divergent trajectories under its action in $G/\Gamma$. We get a sufficient condition for the existence of a non-obvious divergence trajectory in the general case, and a full classification under the assumption that $\mbox{rank}_{\mathbb{Q}}G=\mbox{rank}_{\mathbb{R}}G=2$.