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

Let $H=\mathrm{S}\mathrm{O}(n,1)$ and $A=\{a(t):t\in \mathbb{R}\}$ be a maximal $\mathbb{R}$-split Cartan subgroup of $H$. Let $\mathrm{\Gamma }\subset H\times H$ be a nonuniform lattice in $H\times H$ and $X_{\mathrm{\Gamma }}:=H\times H\mathrm{/}\mathrm{\Gamma }$. Let $A_2:=\{a_2(t):=a(t)\times a(t):t\in \mathbb{R}\}\subset A\times A$ on $X_{\mathrm{\Gamma }}$ and ${\mathcal{D}}_{\mathrm{\Gamma }}\subset X_{\mathrm{\Gamma }}$ denote the collection of points $x\in X_{\mathrm{\Gamma }}$ such that $a_2(t)x$ diverges as $t \rightarrow +\infty$\mathrm{.}Inthisnote,wewillshowthatiftheHausdorffdimensionof$$\mathcal{D}$_{\Gamma}$isgreaterthan$\dim (H\times H) - 2(n-1)$,then$\Gamma $ is essentially split, namely, it contains a subgroup of finite index of form ${\mathrm{\Gamma }}_1\times {\mathrm{\Gamma }}_2$, where ${\mathrm{\Gamma }}_1$ and ${\mathrm{\Gamma }}_2$ are both lattices in $H$.