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.
For each almost simple group G such that S ≤ G ≤ Aut(S) and S is a simple group of order less than 900,000 listed in the Atlas of Finite Groups, we give, up to isomorphism, the number of abstract regular polytopes on which G acts regularly. The results have been obtained using a series of Magma programs. All these polytopes are made available on the first author's website, at
We prove that if G is a string C-group of rank 4 and G ∼ = L 2 (q) with q a prime power, then q must be 11 or 19. The polytopes arising are Grünbaum's 11-cell of type {3, 5, 3} for L 2 (11) and Coxeter's 57-cell of type {5, 3, 5} for L 2 (19), each a locally projective regular 4-polytope.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.