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Non-divergence of unipotent flows on quotients of rank-one semisimple groups
Published online by Cambridge University Press: 28 December 2015
Abstract
Let $G$ be a semisimple Lie group of rank one and
$\unicode[STIX]{x1D6E4}$ be a torsion-free discrete subgroup of
$G$ . We show that in
$G/\unicode[STIX]{x1D6E4}$ , given
$\unicode[STIX]{x1D716}>0$ , any trajectory of a unipotent flow remains in the set of points with injectivity radius larger than
$\unicode[STIX]{x1D6FF}$ for a
$1-\unicode[STIX]{x1D716}$ proportion of the time, for some
$\unicode[STIX]{x1D6FF}>0$ . The result also holds for any finitely generated discrete subgroup
$\unicode[STIX]{x1D6E4}$ and this generalizes Dani’s quantitative non-divergence theorem [On orbits of unipotent flows on homogeneous spaces. Ergod. Th. & Dynam. Sys.4(1) (1984), 25–34] for lattices of rank-one semisimple groups. Furthermore, for a fixed
$\unicode[STIX]{x1D716}>0$ , there exists an injectivity radius
$\unicode[STIX]{x1D6FF}$ such that, for any unipotent trajectory
$\{u_{t}g\unicode[STIX]{x1D6E4}\}_{t\in [0,T]}$ , either it spends at least a
$1-\unicode[STIX]{x1D716}$ proportion of the time in the set with injectivity radius larger than
$\unicode[STIX]{x1D6FF}$ , for all large
$T>0$ , or there exists a
$\{u_{t}\}_{t\in \mathbb{R}}$ -normalized abelian subgroup
$L$ of
$G$ which intersects
$g\unicode[STIX]{x1D6E4}g^{-1}$ in a small covolume lattice. We also extend these results to when
$G$ is the product of rank-one semisimple groups and
$\unicode[STIX]{x1D6E4}$ a discrete subgroup of
$G$ whose projection onto each non-trivial factor is torsion free.
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- © Cambridge University Press, 2015
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