Moebius transformations and the Poincare distance in the quaternionic setting Cinzia BisiGraziano Gentili 30G3530C2030F45functions of hypercomplex variablesquaternionicMoebius transformationsquaternionic Poincare distance and metric In the space $\mathbb{H}$ of quaternions, we investigate the natural, invariant geometry of the open, unit disc $\Delta_{\mathbb{H}}$ and of the open half-space $\mathbb{H}^{+}$. These two domains are diffeomorphic via a Cayley-type transformation. We first study the geometrical structure of the groups of M\"obius transformations of $\Delta_{\mathbb{H}}$ and $\mathbb{H}^{+}$ 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 M\"obius transformations, and use it to define the analog of the Poincar\'e distances and differential metrics on $\Delta_{\mathbb{H}}$ and $\mathbb{H} ^{+}$. Indiana University Mathematics Journal 2009 text pdf 10.1512/iumj.2009.58.3706 10.1512/iumj.2009.58.3706 en Indiana Univ. Math. J. 58 (2009) 2729 - 2764 state-of-the-art mathematics http://iumj.org/access/