Pasini, Antonio Covers of finite geometries with non-spherical minimal circuit diagram. (English) Zbl 0586.51014 Buildings and the geometry of diagrams, Lect. 3rd 1984 Sess. C.I.M.E., Como/Italy 1984, Lect. Notes Math. 1181, 218-241 (1986). [For the entire collection see Zbl 0577.00009.] Truncations of chamber systems are used to show that the universal 2- cover of several classes of finite diagram geometries are infinite. By theorem 1, truncation and forming the universal 2-cover commute under suitable conditions on the chamber system. Theorem 2 provides severe restrictions for a finite rank 3 geometry with finite universal 2-cover [cp. M. A. Ronan, Q. J. Math. Oxf. II. Ser. 32, 225-233 (1981; Zbl 0466.57004)]. Theorems 3, 4 and their corollaries deal with finite regular geometries \(\Gamma\) of rank at least 4 such that \(\Gamma\) is non- spherical over a 3-set of types. If \(\Gamma\) is thick, or if the intersection property holds in \(\Gamma\), then the universal 2-cover of \(\Gamma\) is infinite. The paper ends with a list of 8 examples, where these results are applied to geometries of several (mostly sporadic) finite simple groups. Reviewer: T.Grundhöfer Cited in 7 Documents MSC: 51E30 Other finite incidence structures (geometric aspects) 05B25 Combinatorial aspects of finite geometries 57M15 Relations of low-dimensional topology with graph theory Keywords:building; universal covers of geometries with non-spherical minimal circuit diagram; Truncations of chamber systems; universal 2-cover; diagram geometries Citations:Zbl 0577.00009; Zbl 0466.57004 PDFBibTeX XML