Proceedings of the Indian Academy of Sciences - Mathematical Sciences

, Volume 101, Issue 1, pp 1–17

Asymptotic behaviour of trajectories of unipotent flows on homogeneous spaces

Authors

  • S G Dani
    • School of MathematicsTata Institute of Fundamental Research
  • G A Margulis
    • Institute for Problems of Information Transmission
Article

DOI: 10.1007/BF02872005

Cite this article as:
Dani, S.G. & Margulis, G.A. Proc. Indian Acad. Sci. (Math. Sci.) (1991) 101: 1. doi:10.1007/BF02872005

Abstract

We show that ifG is a semisimple algebraic group defined overQ and Γ is an arithmetic lattice inG:=GR with respect to theQ-structure, then there exists a compact subsetC ofG/Γ such that, for any unipotent one-parameter subgroup {ut} ofG and anyg∈G, the time spent inC by the {ut}-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].

Keywords

Homogeneous spacesunipotent flowstrajectories

Copyright information

© Indian Academy of Sciences 1991