Barré, Sylvain Exotic triangular Tits buildings. (Immeubles de Tits triangulaires exotiques.) (French) Zbl 1003.51007 Ann. Fac. Sci. Toulouse, VI. Sér., Math. 9, No. 4, 575-603 (2000). In this article the author studies the triangular buildings (of type \(\widetilde{A}_2\)), whose vertices, following a theorem of J. Tits [Finite geometries, buildings, and related topics, Pap. Conf., Pingree Park/CO (USA) 1988, 17-28 (1990; Zbl 0744.51013)], are naturally colored by the type of their \(2\)-balls. The new proof of this theorem given by the author allows him to show a prescription result, stating that this coloring can be freely chosen. In the last part of the paper the author studies an exotic triangular building of order \(2\). He firstly proves that in its universal cover the \(2\)-ball of the black vertex is not isomorphic to the one of the white vertex, and then he shows that the fundamental group of this building has finite index (\(6\) or \(12\)) in the one of the universal cover. He provides an example of triangular building of order \(2\) admitting a quotient with two vertices and with isotropy groups of order at most \(12\). Reviewer: Razvan Litcanu (Rennes) Cited in 1 ReviewCited in 9 Documents MSC: 51E24 Buildings and the geometry of diagrams Keywords:triangular buildings; projective plane; 2-ball Citations:Zbl 0744.51013 × Cite Format Result Cite Review PDF Full Text: DOI Numdam EuDML References: [1] Artin, E.). - Algèbre géométrique, Bordas, Paris, 1978. · Zbl 0103.25004 [2] Barré, S.). - Polyèdres finis de dimension 2 à courbure négative ou nulle et de rang 2, Annales de l’Institut Fourier, T45, 1995. · Zbl 0831.53031 [3] Barré, S.). - Polyèdre de rang deux, thèse, ENS Lyon, décembre 1996. 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