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 (Iσ) and the absolute geometry is a generalized hexagon, the moving 5 3 6 absolute geometry also gives interesting flag-transitive geometries with Buekenhout diagram for the groups G2(k) and 3D4(k), for any prime power k ≥ 2. We also classify the absolute geometries for geometries with trialities but no dualities coming from maps of Class III with automorphism group L2(q3), where q ≥ 2 is prime power. We then investigate the moving absolute geometries for these geometries, illustrating their interest in this case.