If we identify the boundary at infinity of the hyperbolic $3$-space $\mathbb{H}^3$ with the complex projective line $\mathbb{P}^1(\mathbb{C})=\mathbb{C} \cup \{ \infty\}$, we know that the ideal tetrahedra $\sigma$ with vertices $(0,\infty,1,\omega)$ is regular, where $\omega=e^{\frac{i\pi}{3}}$. In the group $PO(3,1)$ of isometries of $\mathbb{H}^3$ we can consider the subgroup generated $\Gamma_\sigma = \langle r_0,\ldots,r_3,\mu_2 \rangle$, where each $r_i$ is a reflection along a face of $\sigma$ and $\mu_2$ is the multiplication by two.
Since the subgroup $\Gamma_\sigma$ is dense in $PO(3,1)$, it is known that its natural action by left translation on $PO(3,1)$ is ergodic. Is this action also strongly ergodic?
