Polynomial effective equidistribution for some higher dimensional unipotent subgroups

Abstract: Let G be a semisimple Lie group, Γ be a lattice in G and U be a unipotent subgroup of G. A celebrated theorem of Ratner says that for any x in G/Γ the orbit U.x is equidistributed in a periodic orbit of some subgroup U ≤ L ≤ G. Establishing a quantitative version of Ratner’s theorem has been long sought after. If U is a horospherical subgroup of G, the question is well-studied. If U is not a horospherical subgroup, this question is far less understood. Recently, Lindenstrauss, Mohammadi, Wang and Yang established a fully quantitative and effective equidistribution result for orbits of one-parameter (non-horospherical) unipotent groups in some cases. In this talk, we will discuss a recent equidistribution theorem for some unipotent subgroups in higher dimension. Our results in particular provide equidistribution theorems for orbits of the isometry group of a non-degenerate bilinear form on R^n in SL_n(R)/SL_n(Z).

dynamical systemsnumber theory

Audience: general audience

New England Dynamics and Number Theory Seminar