Abstract Following Poincar\`e's geometric method, we construct two new
nonorientable noncompact hyperbolic space forms by the regular octahedron in
Fig\.~1. The construction is motivated by Thurston's example [6], discussed
also by Apansov [1] in details. Our new space forms will be denoted by
$$
\tilde D_1= H^3/G_1\quad\text{and}\quad \tilde D_2= H^3/G_2,
$$
where $\tilde D_1$ and $\tilde D_2$ are obtained by pairing
faces of $D$ via
isometries of groups $G_1$ and $G_2$, respectively, acting discontinuously and
freely on the hyperbolic 3-space $H^3$ (Fig\.~2, Fig\.~3). These groups are
defined by generators and relations in Sect\.~3. The complete computer
classification of possible space forms by our octahedron will be discussed in
[4], where it turns out that our two space forms are isometric, i.e\. $G_1$
and $G_2$ are conjugated by an isometry $\varphi$ of $H^3$, i.e\.
$G_2=\varphi^{-1}G_1\varphi$,
$$
\align
G_1&=(g_1,g_2,\bar g_1,\bar g_2\;\raise2pt\vbox{\hrule width.5cm}\;g_1\bar
g_1^{-1}g_2\bar g_2^{-1}=g_1g_1g_2g_2=\bar g_1\bar g_1\bar g_2\bar g_2=1),\\
G_2&=(t_1,t_2,\bar g_1,\bar g_2\;\raise2pt\vbox{\hrule width.5cm}\;t_1\bar
g_1^{-1}t_2^{-1}\bar g_2=t_1t_2t_1^{-1}t_2^{-1}=\bar g_1\bar g_1\bar g_2\bar
g_2=1).
\endalign
$$
|