Asymptotic behaviour of trajectories of unipotent flows on homogeneous spaces

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We show that ifG is a semisimple algebraic group defined overQ and Γ is an arithmetic lattice inG:=G R with respect to theQ-structure, then there exists a compact subsetC ofG/Γ such that, for any unipotent one-parameter subgroup {u t} ofG and anyg∈G, the time spent inC by the {u t}-trajectory ofgΓ, during the time interval [0,T], is asymptotic toT, unless {g −1utg} is contained in aQ-parabolic subgroup ofG. Some quantitative versions of this are also proved. The results strengthen similar assertions forSL(n,Z),n≥2, proved earlier in [5] and also enable verification of a technical condition introduced in [7] for lattices inSL(3,R), which was used in our proof of Raghunathan’s conjecture for a class of unipotent flows, in [8].