Using the rings of Lipschitz and Hurwitz integers H(Z) and Hur(Z) in the quaternion division algebra H, we define several Kleinian discrete subgroups of P SL(2, H). We define first a Kleinian subgroup P SL(2, L) of P SL(2, H(Z)). This group is a generalization of the modular group P SL(2, Z). Next we define a discrete subgroup P SL(2, H) of P SL(2, H) which is obtained by using Hurwitz integers and in particular the subgroup of order 24 consisting of Hurwitz units. It contains as a subgroup P SL(2, L). In analogy with the classical modular case, these groups act properly and discontinuously on the hyperbolic half space H 1 H := {q ∈ H : (q) > 0}. We exhibit fundamental domains of the actions of these groups and determine the isotropy groups of the fixed points and describe the orbifold quotients H 1 H /P SL(2, L) and H 1 H /P SL(2, H) which are quaternionic versions of the classical modular orbifold and they are of finite volume. We give a thorough study of the Iwasawa decompositions, affine subgroups, and their descriptions by Lorentz transformations in the Lorentz-Minkowski model of hyperbolic 4-space. We give abstract finite presentations of these modular groups in terms of generators and relations via the Cayley graphs associated to the fundamental domains. We also describe a set of Selberg covers (corresponding to finite-index subgroups acting freely) which are quaternionic hyperbolic manifolds of finite volume with cusps whose sections are 3-tori. These hyperbolic arithmetic 4-manifolds are topologically the complement of linked 2-tori in the 4-sphere, in analogy with the complement in the 3-sphere of the Borromean rings and are related to the ubiquitous hyperbolic 24-cell. Finally we study the Poincaré extensions of these Kleinian groups to arithmetic Kleinian groups acting on hyperbolic 5-space and described in the quaternionic setting. In particular P SL(2, H(Z)) and P SL(2, Hur(Z)) are discrete subgroups of isometries of H 5 R and H 5 R /P SL(2, H(Z)), H 5 R /P SL(2, Hur(Z)) are examples of arithmetic 5-dimensional hyperbolic orbifolds of finite volume.
