Integral binary Hamiltonian forms and their waterworlds

Abstract

We give a graphical theory of integral indefinite binary Hamiltonian forms f f 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 \mathscr {O} in a definite quaternion algebra over Q \mathbb {Q} , we define the waterworld of f f , analogous to Conway’s river and Bestvina-Savin’s ocean, and use it to give a combinatorial description of the values of f f on O × O \mathscr {O}\times \mathscr {O} . We use an appropriate normalisation of Busemann distances to the cusps (with an algebraic description given in an independent appendix), the SL 2 ⁡ ( O ) \operatorname {SL}_{2}(\mathscr {O}) -equivariant Ford-Voronoi cellulation of the real hyperbolic 5 5 -space, and the conformal action of SL 2 ⁡ ( O ) \operatorname {SL}_{2}(\mathscr {O}) on the Hamilton quaternions.