Abstract
AbstractLet G be a semisimple 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 accounts 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\kern-1pt{/}\kern-1pt\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 $\mathrm {rank}_{{\mathbb {Q}}}G=\mathrm {rank}_{{\mathbb {R}}}G=2$ .
Highlights
Let G be a semisimple real algebraic group defined over Q, Ŵ be an arithmetic subgroup of G, and A ⊂ G be a semigroup
Some sets of exceptional trajectories are related to classical problems in number theory
Let G be a semisimple real algebraic group defined over Q and Ŵ be an arithmetic subgroup of G
Summary
Let G be a semisimple real algebraic group defined over Q, Ŵ be an arithmetic subgroup of G, and A ⊂ G be a semigroup. Our main result provides a necessary condition for the existence of non-obvious divergence trajectories under the action of closed cones in T on G/Ŵ. Our second result provides a sufficient condition for the existence of a non-obvious divergence trajectory under the action of closed cones in T on G/Ŵ. It shows that when rankQG = rankRG = 2, any closed cone which does not satisfy the assumption of Theorem 1.6 does not satisfy its conclusion as well. For any unbounded closed cone A ⊂ Aǫ, there exists a non-obvious divergent trajectory for the action of A on G/Ŵ.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
