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New time-changes of unipotent flows on quotients of Lorentz groups

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  • 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.

    Mathematics Subject Classification: Primary: 37A17; Secondary: 37A20.


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  • Figure 1.  A collection of $ \epsilon $-blocks $ \{{\rm{BL}}_{1}, \ldots, {\rm{BL}}_{n}\} $. The solid straight lines are the unipotent orbits in the $ \epsilon $-blocks and the dashed lines are the rest of the unipotent orbits. The bent curves indicate the length defined by the letters

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