  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
$\Gamma_1\backslash G/\Gamma_2$, 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, $\Gamma_1$ and $\Gamma_2$ are discrete subgroups of $G$ and
$\Gamma_2$ is cyclic. A construction of polyhedral fundamental domains for the
action of $\Gamma_1\times\Gamma_2$ on $G$ via $(g,h)\cdot x=gxh^{-1}$ 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 $\Gamma_1\backslash G/\Gamma_2$ 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.
