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Non-Divergence of Unipotent Flows on Quotients of Rank One Semisimple Groups

Abstract

Let GG be a semisimple Lie group of rank 11 and Γ\Gamma be a torsion free discrete subgroup of GG. We show that in G/ΓG/\Gamma, given ϵ>0\epsilon>0, any trajectory of a unipotent flow remains in the set of points with injectivity radius larger than δ \delta for 1ϵ1-\epsilon proportion of the time for some δ>0\delta>0. The result also holds for any finitely generated discrete subgroup Γ\Gamma and this generalizes Dani's quantitative nondivergence theorem \cite{D} for lattices of rank one semisimple groups. Furthermore, for a fixed ϵ>0\epsilon>0 there exists an injectivity radius δ\delta such that for any unipotent trajectory {utx}t[0,T]\{u_tx\}_{t\in [0,T]}, either it spends at least 1ϵ1-\epsilon proportion of the time in the set with injectivity radius larger than δ\delta for all large T>0T>0 or there exists a {ut}tR\{u_t\}_{t\in\mathbb{R}}-normalized abelian subgroup LL of GG which intersects gΓg1g\Gamma g^{-1} in a small covolume lattice. We also extend these results when GG is the product of rank-11 semisimple groups and Γ\Gamma a discrete subgroup of GG whose projection onto each nontrivial factor is torsion free.Comment: 23 page

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