Ronan, M. A. Triangle geometries. (English) Zbl 0568.51002 J. Comb. Theory, Ser. A 37, 294-319 (1984). The aim of this paper is to study triangle geometries. A triangle geometry is a rank 3 incidence geometry or, using the recent concept of a chamber system, a rank 3 chamber system whose rank 2 residues are projective planes. (By a theorem of Tits, every triangle geometry arises as quotient of a suitable affine building). The author investigates two special classes of triangle geometries: the trivalent ones (whose rank 1 residues have three chambers) and the tight ones (containing exactly three rank 2 residues). For the first class he shows that a regular automorphism group of a trivalent geometry can only be of four types and gives an example for three of these types. For the second class the author gives a method of construction using projective planes; in particular he proves that any finite desarguesian plane gives rise to a tight triangle geometry which admits the full Singer group of the plane, together with its multipliers. Special types of tight triangle geometries are studied, too. Reviewer: C.Bartolone Cited in 2 ReviewsCited in 17 Documents MSC: 51A05 General theory of linear incidence geometry and projective geometries 51D15 Abstract geometries with parallelism Keywords:trivalent triangle geometries; tight triangle geometries; triangle geometry; affine building × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Bruhat, F.; Tits, J., Groupes réductifs sur un corps local. I. Données radicielles valuées, Publ. Math. Inst. Hautes Etudes Sci., 41, 5-251 (1972) · Zbl 0254.14017 [2] Chermak, A., On certain groups with parabolic-type subgroups over \(Z_2\), J. London Math. Soc. (2), 265-279 (1981) · Zbl 0464.20018 [3] M. HallDuke Math. J.14; M. HallDuke Math. J.14 · Zbl 0029.22502 [4] Higman, D. G.; McLaughlin, J. E., Geometric ABA groups, Illinois J. Math., 5, 382-397 (1961) · Zbl 0104.14702 [5] Hughes, D. R.; Piper, F. C., Projective Planes (1973), Springer-Verlag: Springer-Verlag Berlin/Heidelberg/New York · Zbl 0267.50018 [6] P. Köhler, Th. Meixner, and M. Wester; P. Köhler, Th. Meixner, and M. Wester [7] Lyndon, R.; Schupp, P., Combinatorial Group Theory (1977), Springer-Verlag: Springer-Verlag New York/Berlin · Zbl 0368.20023 [8] Niles, R., Finite groups with parabolic-type subgroups must have a BN pair, J. Algebra, 75, 484-494 (1982) · Zbl 0491.20020 [9] U. Ott; U. Ott [10] Ronan, M. A., Coverings and automorphisms of chamber systems, European J. Combin., 1, 259-269 (1980) · Zbl 0553.51004 [11] Serre, J.-P, Cohomologie des Groupes Discrets, (Prospects in Mathematics. Prospects in Mathematics, Annals of Math Studies 70 (1971), Princeton Univ. Press: Princeton Univ. Press Princeton, N.J), 77-169 · Zbl 0229.57016 [12] Singer, J., A theorem in finite projective geometry and some applications to number theory, Trans. Amer. Math. Soc., 43, 377-385 (1938) · JFM 64.0972.04 [13] Spanier, E., Algebraic Topology (1966), McGraw-Hill: McGraw-Hill New York · Zbl 0145.43303 [14] F. G. Timmesfeld; F. G. Timmesfeld · Zbl 0521.20007 [15] Tits, J., Buildings of Spherical Type and Finite BN-Pairs, (Lecture Notes in Mathematics No. 386 (1974), Springer-Verlag: Springer-Verlag New York/Berlin) · Zbl 0295.20047 [16] Tits, J., A local approach to buildings, (The Geometric Vein (Coxeter Festschrift) (1981), Springer-Verlag: Springer-Verlag New York/Berlin), 519-547 · Zbl 0496.51001 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.