  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.
