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 $t\to +\infty$. 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 $H$.

arXiv comment6 pages. The paper is withdrawn due to an error in the proof

References in corpus (1)

• Hausdorff dimension of divergent diagonal geodesics on product of finite volume hyperbolic spaces