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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

MSC:

51E30 Other finite incidence structures (geometric aspects)
05B25 Combinatorial aspects of finite geometries
57M15 Relations of low-dimensional topology with graph theory