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

TL;DR

This paper investigates special lattices in Lorentz groups where the Laplace-Beltrami operator has eigenvalues in (0,1/4), and demonstrates the existence of non-measurably conjugate time-changed unipotent flows on these quotients.

Contribution

It introduces new time-changes of unipotent flows on Lorentz group quotients that are not measurably conjugate to original flows, utilizing advanced branching of the complementary series.

Findings

Existence of non-measurably conjugate time-changed flows.

Identification of specific lattices with eigenvalues in (0,1/4).

Enhanced understanding of unipotent flow dynamics on Lorentz quotients.

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.