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 $\phi: I=[a,b] \rightarrow H$, we will show that if $\phi$ satisfies certain geometric condition, then for a typical diagonal subgroup $A ={a(t): t \in \mathbb{R}} \subset H$ the translates ${a(t)\phi(I)x: t >0}$ of the curve $\phi(I)x$ will tend to be equidistributed in $X$ as $t \rightarrow +\infty$. The proof is based on the study of linear representations of $\mathrm{SO}(n,1)$ and $H$.