Highly symmetric hypertopes (Q329406)

Regular polytopes can be described as thin residually connected regular geometries with string diagrams. In the paper under review, the authors generalize the notion of polytope and introduce hypertopes as thin residually connected incidence geometries. Regular (ie, flag-transitive) hypertopes are described as coset geometries and the automorphism group of a regular hypertope is shown to be a smooth quotient of a Coxeter group. On the other hand, not every \(C\)-group gives rise to a hypertope as is shown by examples of regular hypertopes of rank 3 that are not polytopes. Moreover, the authors extend the notion of chirality in polytopes to hypertopes, and in fact to thin incidence geometries. Chiral hypertopes again can be obtained as coset geometries and a characterisation of their automorphism groups is given. This leads to the notion of \(C^+\)-group. Examples are rotation subgroups of \(C\)-groups if they have index 2.