Kakutani Equivalence of Unipotent Flows

Abstract

We study Kakutani equivalence in the class of unipotent flows acting on finite volume quotients of semisimple Lie groups. For every such flow we compute the Kakutani invariant of M. Ratner, the value of which being explicitly given by the Jordan block structure of the unipotent element generating the flow. This in particular answers a question of M. Ratner. Moreover it follows that the only standard unipotent flows are given by $\left(\begin{array}{cc}1& t\\ 0& 1\end{array}\right)\times \mathrm{id}$ acting on $(SL(2,\mathbb{R})\times G^{\prime })/{\Gamma }^{\prime }$, where ${\Gamma }^{\prime }$ is an irreducible lattice in $SL(2,\mathbb{R})\times G^{\prime }$ (with the possibility that $G^{\prime }=\{e\}$).

arXiv commentThe first version of this paper only addresses the case of compact homogeneous spaces. The updated version includes extra arguments to control behavior of orbits which enter the cusps

References in corpus (1)

• Introduction to Arithmetic Groups

Cited by in corpus (5)

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• Parabolic perturbations of unipotent flows on compact quotients of $\text{SL}(3,\mathbb{R})$