Octahedral noncompact hyperbolic space forms with finite volume

Abstract

Following Poincarè'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\. $G_2=\varphi^{-1}G_1\varphi$,$$\alignG_1&=(g_1,g_2,\bar g_1,\bar g_2\;\raise2pt\vbox{\hrule width.5cm}\;g_1\bar g_1^{-1}g_2\bar g_2^{-1}=g_1g_1g_2g_2=\bar g_1\bar g_1\bar g_2\bar g_2=1), G_2&=(t_1,t_2,\bar g_1,\bar g_2\;\raise2pt\vbox{\hrule width.5cm}\;t_1\bar g_1^{-1}t_2^{-1}\bar g_2=t_1t_2t_1^{-1}t_2^{-1}=\bar g_1\bar g_1\bar g_2\bar g_2=1).\endalign$$