Effective Equidistribution of Translates of Large Submanifolds in Semisimple Homogeneous Spaces

Abstract

Let |$G=SL_2(\mathbb{R})^d$| and |$\Gamma=\Gamma_0^d$| with |$\Gamma_0$| a lattice in |$SL_2(\mathbb{R})$|⁠. Let |$S$| be any “curved” submanifold of small codimension of a maximal horospherical subgroup of |$G$| relative to an |$\mathbb{R}$|-diagonalizable element |$a$| in the diagonal of |$G$|⁠. Then for |$S$| compact our result can be described by saying that |$a^n \text{vol}_S$| converges in an effective way to the volume measure of |$G/\Gamma$| when |$n\to \infty$|⁠, with |$\text{vol}_S$| the volume measure on |$S$|⁠.