On certain geometries of type \(C_ n\) and \(F_ 4\)
The author studies the conditions of the geometries of type \(C_ n\) or \(F_ 4\) to be buildings. Let \(\Gamma\) be a geometry of type \(C_ n\) with uniform order \(q(>1)\). When \(n\geq 4\), \(\Gamma\) is a building if \(\Gamma\) is regular \((n=4)\) or all residues in \(\Gamma\) of type \(C_ 4\) are regular \((n>4)\). Hence, if the automorphism group of \(\Gamma\) is transitive on the set of elements of type n-4, then \(\Gamma\) is a building. It is said that the property (LL) holds in \(\Gamma\) if for two distinct points there is at most one line through them, which was considered by \textit{J. Tits} [The geometric vein, The Coxeter Festschr., 519-547 (1981; Zbl 0496.51001)]. As a characterization of buildings, the author proves that \(\Gamma\) is a building iff the property (LL) holds in \(\Gamma\) \((n=3\) or 4), or in all residues in \(\Gamma\) of type \(C_ 4\) (n\(\geq 5)\). If \(n=3\) or 4, then \(\Gamma\) is 2-connected, that is, \(\Gamma\) is its own universal 2-cover. When \(n\geq 5\), \(\Gamma\) is 2- connected if the property (LL) holds in \(\Gamma\) and in all residues in \(\Gamma\) of type \(C_ m\) for every \(m=5,6,...,n-1\). Similar results are also obtained for a geometry in \((F_ 4)_ q\) with uniform order \(q(>1)\).
scientific article; zbMATH DE number 3953685