We study incidence geometries that are thin and residually connected. These geometries generalise abstract polytopes. In this generalised setting, guided by the ideas from the polytopes theory, we introduce the concept of chirality, a property of orderly asymmetry occurring frequently in nature as a natural phenomenon. The main result in this paper is that automorphism groups of regular and chiral thin residually connected geometries need to be C-groups in the regular case and C + -groups in the chiral case.

No abstract

Abstract. If Γ is a string C-group which is isomorphic to a transitive subgroup of the symmetric group S n (other than S n and the alternating group A n ), then the rank of Γ is at most n/2 + 1, with finitely many exceptions (which are classified). It is conjectured that only the symmetric group has to be excluded.

No abstract

No abstract