Divergent trajectories under diagonal geodesic flow and splitting of discrete subgroups of $\mathrm{SO}(n,1) \times \mathrm{SO}(n,1)$

Abstract

Let $H = \mathrm{SO}(n,1)$ and $A = \{a(t): t \in \mathbb{R}\}$ be a maximal $\mathbb{R}$-split Cartan subgroup of $H$. Let $\Gamma \subset H \times H$ be a nonuniform lattice in $H \times H$ and $X_{\Gamma} : = H \times H/ \Gamma$. Let $A_2 : = \{ a_2(t):=a(t) \times a(t) : t \in \mathbb{R}\} \subset A\times A$ on $X_{\Gamma}$ and $\mathcal{D}_{\Gamma}\subset X_{\Gamma}$ denote the collection of points $x \in X_{\Gamma}$ such that $a_2(t)x$ diverges as $t \rightarrow +\infty$. In this note, we will show that if the Hausdorff dimension of $\mathcal{D}_{\Gamma}$ is greater than $\dim (H\times H) - 2(n-1)$, then $\Gamma $ is essentially split, namely, it contains a subgroup of finite index of form $\Gamma_1 \times \Gamma_2 $, where $\Gamma_1$ and $\Gamma_2$ are both lattices in $H$.