Geometries admitting trialities for the symmetric and alternating groups

In incidence geometry, a triality is a symmetry cyclically exchanging triples of types of elements. Requiring geometries with trialities to satisfy standard regularity conditions makes their construction highly non trivial, and known examples are rare. In this paper, we present two infinite families of flag transitive, thin and residually connected geometries admitting trialities and no dualities with type preserving automorphism groups isomorphic to Sym(n) or Alt(n). We also develop general methods that extend to other settings. The residues of the two families are fundamentally different. In the first family, the maximal parabolics have constant size while in the other their size grows essentially linearly with the degree of the group.