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

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Let G be a semisimple Lie group of rank 1 and Γ be a torsion free discrete subgroup of G. We show that in G/Γ, given ε>0, any trajectory of a unipotent flow remains in the set of points with injectivity radius larger than δ for 1-ε proportion of the time for some δ>0. The result also holds for any finitely generated discrete subgroup Γ and this generalizes Dani's quantitative nondivergence theorem D for lattices of rank one semisimple groups. Furthermore, for a fixed ε>0 there exists an injectivity radius δ such that for any unipotent trajectory {uₜx}ₜ∈ [₀,ₜ], either it spends at least 1-ε proportion of the time in the set with injectivity radius larger than δ for all large T>0 or there exists a {uₜ}t-normalized abelian subgroup L of G which intersects gΓg⁻¹ in a small covolume lattice. We also extend these results when G is the product of rank-1 semisimple groups and Γ a discrete subgroup of G whose projection onto each nontrivial factor is torsion free.