Effective equidistribution of unipotent orbits in homogeneous spaces of $\SL(2,\R)\ltimes(\R^2)^{k}$

Abstract

Let $G=\SL(2,\R)\ltimes(\R^2)^{k}$, let $\Gamma$ be a congruence subgroup of $\SL(2,\Z)\ltimes(\Z^2)^{k}$, and let $u_{\mathbb{R}}=(u_x{)}_{x\in \mathbb{R}}$ be the one-parameter subgroup of $G$ given by $u_x=\left(\matr 1x01,0\right)$. We prove polynomially effective asymptotic equidistribution results for expanding translates of $u_{\mathbb{R}}$-orbits and for long pieces of individual $u_{\mathbb{R}}$-orbits in $\Gamma \backslash G$. An important ingredient of the proof is the delta symbol version of the circle method.