The combinatorial geometry of singularities and Arnold's series $E$, $Z$, $Q$

We consider discrete subgroups $\Gamma$ of the simply connected Lie group $\widetilde{\rm SU}(1,1)$ of finite level. This Lie group has the structure of a 3-dimensional Lorentz manifold coming from the Killing form. $\Gamma$ acts on $\widetilde{\rm SU}(1,1)$ by left translations. We want to describe the Lorentz space form $\Gamma\setminus\widetilde{\rm SU}(1,1)$ by constructing a fundamental domain $F$ for $\Gamma$. We want $F$ to be a polyhedron with totally geodesic faces. We construct such $F$ for all $\Gamma$ satisfying the following condition: The image $\overline\Gamma$ of $\Gamma$ in ${\rm PSU}(1,1)$ has a fixed point $u$ in the unit disk of order larger than the level of $\Gamma$. The construction depends on $\Gamma$ and $\Gamma u$.
For co-compact ${\rm\Gamma}$ the Lorentz space form $\Gamma\setminus\widetilde{\rm SU}(1,1)$ is the link of a quasi-homogeneous Gorenstein singularity. The quasi-homogeneous singularities of Arnold's series $E$, $Z$, $Q$ are of this type. We compute the fundamental domains for the corresponding group. They are represented by polyhedra in Lorentz 3-space shown on Tables 1–13. Each series exhibits a regular characteristic pattern of its combinatorial geometry related to classical uniform polyhedra.