Regular Incidence Complexes, Polytopes, and C-Groups

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.

Locally spherical hypertopes from generalised cubes

“…Nevertheless, Schulte was not aware of this. In [29] he gives a nice historical note on the development of the theory.…”

Locally spherical hypertopes from generlised cubes

“…This result was first proved in [19,21] for so-called regular incidence complexes, (combinatorial objects slightly more general than abstract polytopes). See [23,Section 8] for some historical notes on the subject. Analogously, it is also possible to construct, under certain conditions, a regular hypertope from a group, and particularly from a C-group, using the following proposition.…”