Regular Incidence Complexes, Polytopes, and C-Groups

TL;DR

This paper explores the structure and symmetry groups of regular incidence complexes, generalizing concepts from regular polytopes and classifying groups acting transitively on these complexes.

Contribution

It characterizes automorphism groups of regular incidence complexes as generalized string C-groups and studies extensions and special cases close to abstract polytopes.

Findings

Automorphism groups are characterized as generalized string C-groups.

Extensions of regular incidence complexes are systematically studied.

Certain complexes close to polytopes are specifically investigated.

Abstract

Regular incidence complexes are combinatorial incidence structures generalizing regular convex polytopes, regular complex polytopes, various types of incidence geometries, and many other highly symmetric objects. The special case of abstract regular polytopes has been well-studied. The paper describes the combinatorial structure of a regular incidence complex in terms of a system of distinguished generating subgroups of its automorphism group or a flag-transitive subgroup. Then the groups admitting a flag-transitive action on an incidence complex are characterized as generalized string C-groups. Further, extensions of regular incidence complexes are studied, and certain incidence complexes particularly close to abstract polytopes, called abstract polytope complexes, are investigated.