On trialities and their absolute geometries

Abstract

We introduce the notion of moving absolute geometry of a geometry with triality and show that, in the classical case where the triality is of type $(I_{\sigma })$ and the absolute geometry is a generalized hexagon, the moving absolute geometry also gives interesting flag-transitive geometries with Buekenhout diagram with parameters $(d_p,g,d_L)=(5,3,6)$ for the groups $G_2(k)$ and ${}^3{D}_4(k)$, for any integer $k\ge 2$. We also classify the classical absolute geometries for geometries with trialities but no dualities coming from maps of Class III with automorphism group $L_2(q^3)$, where $q$ is a power of a prime. We then investigate the moving absolute geometries for these geometries, illustrating their interest in this case.