Regular Incidence Complexes, Polytopes, and C-Groups
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.
KeywordsAbstract polytope Regular polytope C-group Incidence geometries
MSC 2010Primary: 51M20 Secondary: 52B15 51E24
I am grateful to the referees for their careful reading of the original manuscript and their helpful suggestions that have improved this article.
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