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Finite \(C_ n\) geometries: A survey. (English) Zbl 0753.51008
This paper is an excellent survey of about everything what is known concerning the classification of residual connected Buekenhout-Tits geometries with Buekenhout diagram \(C_ n\). A complete classification of such geometries is hopeless, but with some additional hypotheses, large subclasses can be classified. The additional hypotheses considered in the paper under review are combinations of: being ordinary (= all lines are thick), flat (= all hyperlines are incident with all elements not being hyperlines nor hyperplanes), degenerate (= non-ordinary), thick, thin; admitting parameters of known type, classical type, semi-classical type; being locally classical, finite, infinite; admitting a flag-transitive automorphism group. The paper is roughly divided in two pieces: one part (section 1) describes the known results and states the theorems, giving useful comments and conjectures to let the reader feel where the theory is coming from and where the research is going to. In the second part (sections 2, 3, 4, 5), sketches of the proofs are given. The authors have tried to unify the theory and consequently they were able to simplify existing proofs. Also background information is not avoided, e.g., results on flag-transitive projective planes and generalized quadrangles is given in full generality.
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51E24 Buildings and the geometry of diagrams
51-02 Research exposition (monographs, survey articles) pertaining to geometry