Quaternionic Kleinian modular groups and arithmetic hyperbolic orbifolds over the quaternions

Abstract

Using the rings of Lipschitz and Hurwitz integers $$\mathbb {H}(\mathbb {Z})$$ and $$\mathbb {H}ur(\mathbb {Z})$$ in the quaternion division algebra $$\mathbb {H}$$ , we define several Kleinian discrete subgroups of $$PSL(2,\mathbb {H})$$ . We define first a Kleinian subgroup $$PSL(2,\mathfrak {L})$$ of $$PSL(2,\mathbb {H}(\mathbb {Z}))$$ . This group is a generalization of the modular group $$PSL(2,\mathbb {Z})$$ . Next we define a discrete subgroup $$PSL(2,\mathfrak {H})$$ of $$PSL(2,\mathbb {H})$$ which is obtained by using Hurwitz integers. It contains as a subgroup $$PSL(2,\mathfrak {L})$$ . In analogy with the classical modular case, these groups act properly and discontinuously on the hyperbolic quaternionic half space. We exhibit fundamental domains of the actions of these groups and determine the isotropy groups of the fixed points and describe the orbifold quotients $$\mathbf {H}_{\mathbb {H}}^1/PSL(2,\mathfrak {L})$$ and $$\mathbf {H}_{\mathbb {H}}^1/PSL(2,\mathfrak {H})$$ which are quaternionic versions of the classical modular orbifold and they are of finite volume. Finally we give a thorough study of their descriptions by Lorentz transformations in the Lorentz–Minkowski model of hyperbolic 4-space.