Two series of polyhedral fundamental domains for Lorentz bi-quotients

The main aim of this paper is to give two infinite series of examples of Lorentz space forms that can be obtained from Lorentz polyhedra by identification of faces. These Lorentz space forms are bi-quotients of the form ${\mathrm{\Gamma }}_1\mathrm{\backslash }G\mathrm{/}{\mathrm{\Gamma }}_2$, where $G=\widetilde{\operatorname{SU}(1,1)}\cong\widetilde{\operatorname{SL}(2,{\mathbb R})}$ is a simply connected Lie group with the Lorentz metric given by the Killing form, ${\mathrm{\Gamma }}_1$ and ${\mathrm{\Gamma }}_2$ are discrete subgroups of $G$ and ${\mathrm{\Gamma }}_2$ is cyclic. A construction of polyhedral fundamental domains for the action of ${\mathrm{\Gamma }}_1\times {\mathrm{\Gamma }}_2$ on $G$ via $(g,h)\cdot x=gx{h}^{-1}$ was given in the earlier work of the second author. In this paper we give an explicit description of the fundamental domains obtained by this construction for two infinite series of groups. These results are connected to singularity theory as the bi-quotients ${\mathrm{\Gamma }}_1\mathrm{\backslash }G\mathrm{/}{\mathrm{\Gamma }}_2$ appear as links of certain quasi-homogeneous $\mathbb{Q}$-Gorenstein surface singularities, i.e.\ the intersections of the singular variety with sufficiently small spheres around the isolated singular point.