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 $S$.