Existence of Non-Obvious Divergent Trajectories in homogeneous spaces

DOI10.1007/S11856-021-2274-2arXiv1909.09205MaRDI QIDQ6325649

Nattalie Tamam

Publication date: 19 September 2019

Abstract: We prove a modified version for a conjecture of Weiss from 2004. Let $G$ be a semisimple real algebraic group defined over $mathbbQ$, $Gamma$ 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 $mathbbQ$-factor of $G$ is of small enough dimension (relatively to the $mathbbQ$-rank of the $mathbbQ$-factor), then there are non-obvious divergent trajectories for the action of $A$ on $G/Gamma$.

Mathematics Subject Classification ID

Discrete subgroups of Lie groups (22E40) Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.) (37D40) General groups of measure-preserving transformations and dynamical systems (37A15)

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