Equidistribution of expanding translates of curves in homogeneous spaces with the action of $(\mathrm{SO}(n,1){)}^k$

Abstract

Given a homogeneous space $X=G/\Gamma$ with $G$ containing the group $H=(\mathrm{SO}(n,1){)}^k$. Let $x\in X$ such that $Hx$ is dense in $X$. Given an analytic curve $\varphi :I=[a,b]\to H$, we will show that if $\varphi$ satisfies certain geometric condition, then for a typical diagonal subgroup $A=\{a(t):t\in \mathbb{R}\}\subset H$ the translates $\{a(t)\varphi (I)x:t>0\}$ of the curve $\varphi (I)x$ will tend to be equidistributed in $X$ as $t\to +\infty$. The proof is based on the study of linear representations of $\mathrm{SO}(n,1)$ and $H$.

19 pages