Moebius transformations and the Poincare' distance in the quaternionic setting

Abstract. In the space H of quaternions, we investigate the natural, invariant geometry of the open, unit disc Delta_H and of the open half-space H^+. These two domains are diffeomorphic via a Cayley-type transformation. We first study the geometrical structure of the groups of Moebius transformations of Delta_H and H^+ and identify original ways of representing them in terms of two (isomorphic) groups of matrices with quaternionic entries. We then define the cross-ratio of four quaternions, prove that, when real, it is invariant under the action of the Moebius transformations, and use it to define the analogous of the Poincare' distances and differential metrics on Delta_H and H^+.