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

Abstract

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 $(0,1/4)$, 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.