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Zbl 1003.51007
Barré, Sylvain
Exotic triangular Tits buildings. (Immeubles de Tits triangulaires exotiques.) (French)
[J] Ann. Fac. Sci. Toulouse, VI. Sér., Math. 9, No.4, 575-603 (2000). ISSN 0240-2963

In this article the author studies the triangular buildings (of type $\widetilde{A}_2$), whose vertices, following a theorem of {\it 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. \par 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 $12$.
[Razvan Litcanu (Rennes)]

MSC 2000:: *51E24 Buildings and the geometry of diagrams

Keywords: triangular buildings; projective plane; 2-ball

Citations: Zbl 0744.51013

Cited in: Zbl 1142.51011

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