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

Abstract

We study the cocompact lattices \begin{document}$ \Gamma\subset SO(n, 1) $\end{document} so that the Laplace–Beltrami operator \begin{document}$ \Delta $\end{document} on \begin{document}$ SO(n)\backslash SO(n, 1)/\Gamma $\end{document} has eigenvalues in \begin{document}$ (0, \frac{1}{4}) $\end{document}, and then show that there exist time-changes of unipotent flows on \begin{document}$ SO(n, 1)/\Gamma $\end{document} 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.