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

Abstract

<p style='text-indent:20px;'>We study the cocompact lattices <inline-formula><tex-math id="M1">\begin{document}$ \Gamma\subset SO(n, 1) $\end{document}</tex-math></inline-formula> so that the Laplace–Beltrami operator <inline-formula><tex-math id="M2">\begin{document}$ \Delta $\end{document}</tex-math></inline-formula> on <inline-formula><tex-math id="M3">\begin{document}$ SO(n)\backslash SO(n, 1)/\Gamma $\end{document}</tex-math></inline-formula> has eigenvalues in <inline-formula><tex-math id="M4">\begin{document}$ (0, \frac{1}{4}) $\end{document}</tex-math></inline-formula>, and then show that there exist time-changes of unipotent flows on <inline-formula><tex-math id="M5">\begin{document}$ SO(n, 1)/\Gamma $\end{document}</tex-math></inline-formula> 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.</p>