Integral binary Hamiltonian forms and their waterworlds.

We give a graphical theory of integral indefinite binary Hamiltonian forms ƒ analogous to the one of Conway for binary quadratic forms and the one of Bestvina-Savin for binary Hermitian forms. Given a maximal order O in a definite quaternion algebra over Q, we define the waterworld of ƒ, analogous to Conway's river and Bestvina-Savin's ocean , and use it to give a combinatorial description of the values of ƒ on O × O. We use an appropriate normalisation of Busemann distances to the cusps (with an algebraic description given in an independent appendix), the SL₂(O)-equivariant Ford-Voronoi cellulation of the real hyperbolic 5-space, and the conformal action of SL₂(O) on the Hamilton quaternions. [ABSTRACT FROM AUTHOR]