Abstract: We introduce the notion of moving absolute geometry of a geometry with triality and show that certain cases the moving absolute geometry of the classical 7-dimensional quadric also gives interesting flag-transitive geometries. We also classify the classical absolute geometries for geometries with trialities but no dualities coming from maps (graphs embeddings on surfaces) of Class III with automorphism group L_2(q_3), where q is a power of a prime. This is joint work with Dimitri Leemans and Klara Stokes.
