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 $Γ\subset H \times H$ be a nonuniform lattice in $H \times H$ and $X_Γ : = H \times H/ Γ$. Let $A_2 : = \{ a_2(t):=a(t) \times a(t) : t \in \mathbb{R}\} \subset A\times A$ on $X_Γ$ and $\mathcal{D}_Γ\subset X_Γ$ denote the collection of points $x \in X_Γ$ such that $a_2(t)x$ diverges as $t \rightarrow +\infty$. In this note, we will show that if the Hausdorff dimension of $\mathcal{D}_Γ$ is greater than $\dim (H\times H) - 2(n-1)$, then $Γ$ is essentially split, namely, it contains a subgroup of finite index of form $Γ_1 \times Γ_2 $, where $Γ_1$ and $Γ_2$ are both lattices in $H$.