Reflection groups on the octave hyperbolic plane (Q1282344)

The paper presents three examples of discrete reflection groups. Two of them act with finite covolume on the octave hyperbolic plane \(\mathbb{Q} H^2\), the third one is a discrete group of isometries of the octave hyperbolic line \(\mathbb{O} H^1\). After having introduced the algebra \(\mathbb{Q}\) of octaves the author defines the octave projective plane \(\mathbb{Q} P^2\) and the octave hyperbolic plane \(\mathbb{Q} H^2\) by using the Jordan algebra approach of \textit{H. Freudenthal} [Geom. Dedicata 19, 7-63 (1985; Zbl 0573.51004)]. He considers two reflections \(R\), \(R'\). The discrete groups of isometries of the plane \(\mathbb{Q} H^2\) will be generated by conjugates of \(R\) \(R'\). They are subgroups of the Lie group \(\Aut\mathbb{Q} H^2\). The discrete subgroup of isometries of \(\mathbb{Q} H^1\) is constructed by using the fact that \(\mathbb{Q} H^1\) is isomorphic to the real hyperbolic space \(H^8\).