We study the cocompact lattices $ \Gamma\subset SO(n, 1) $ so that the Laplace–Beltrami operator $ \Delta $ on $ SO(n)\backslash SO(n, 1)/\Gamma $ has eigenvalues in $ (0, \frac{1}{4}) $, and then show that there exist time-changes of unipotent flows on $ SO(n, 1)/\Gamma $ that are not measurably conjugate to the unperturbed ones. A main ingredient of the proof is a stronger version of the branching of the complementary series. Combining it with a refinement of the works of Ratner and Flaminio–Forni is adequate for our purpose.
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