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Divergent trajectories in arithmetic homogeneous spaces of rational rank two
Published online by Cambridge University Press: 05 November 2020
Abstract
Let 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
$\mathrm {rank}_{{\mathbb {Q}}}G=\mathrm {rank}_{{\mathbb {R}}}G=2$
.
MSC classification
- Type
- Original Article
- Information
- Creative Commons
- This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
- Copyright
- © The Author(s), 2020. Published by Cambridge University Press
References
REFERENCES
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