4-transitivity and cross-ratio in Moufang-Klingenberg planes over rings of plural numbers
It is proved that the collineation group \(\mathcal{G}\) in the Moufang-Klingenberg plane \(\mathbb{M}(\mathcal{A})\), with \(\mathcal{A}\) the local ring of plural numbers \(\mathcal{A}:=\mathbb{O}[\eta]/\langle \eta^m \rangle\) over a Cayley division algebra \(\mathbb{O}\), acts transitively on the set of 4-gons. Also the cross-ratio of four distinct points on a line in \(\mathbb{M}(\mathcal{A})\) is discussed, inspired by work of \textit{A. Blunck} [Result. Math. 19, No 3--4, 246--256 (1991; Zbl 0725.51001); Geom. Dedicata 43, No 1, 93--107 (1992; Zbl 0758.51005)].
scientific article; zbMATH DE number 7344133
scientific article