On Topologically Big Divergent Trajectories

TL;DR

This paper proves that divergent trajectories of certain algebraic group actions on homogeneous spaces diverge for purely algebraic reasons, and explores orbit intersections, solving longstanding conjectures in the field.

Contribution

It establishes that all divergent $A$-orbits diverge algebraically and demonstrates that $A$-orbits intersect deformation retracts, resolving conjectures by Weiss, Pettet, and Souto.

Findings

Divergent $A$-orbits diverge algebraically.

Every $A$-orbit intersects deformation retracts in many cases.

Solves longstanding conjectures on orbit divergence and intersection.

Abstract

We study the behavior of $A$-orbits in $G/\Gamma$, when $G$ is a semisimple real algebraic $\mathbb{Q}$-group, $\Gamma$ is a non-uniform arithmetic lattice, and $A$ is a torus of dimension $\geq\operatorname{rank}_\mathbb{Q}(\Gamma)$. We show that every divergent trajectory of $A$ diverges due to a purely algebraic reason, % has a simple algebraic description. solving a longlasting conjecture of Weiss. In addition, we examine the intersections of $A$-orbits and show that in many cases every $A$-orbit intersects every deformation retract $X\subseteq G/\Gamma$. This solves the questions raised by Pettet and Souto. The proofs use algebraic and differential topology, as well as algebraic group theory.