Arborescent constructions of buildings
\textit{J. Tits} classified the locally finite buildings of spherical type [`Buildings of spherical type and finite BN-pairs', Lect. Notes Math. 386, Springer-Verlag (1974; Zbl 0295.20047)] and those of Euclidean type [Lect. Notes Math. 1181, 159-190 (1986; Zbl 0611.20026)]. In both cases the set of the isomorphism classes of the buildings is countable. For buildings of hyperbolic type we have a quite different situation. A general classification theorem is not known, only some special results have been obtained. For the two-dimensional case an uncountable set of locally finite hyperbolic buildings was constructed by \textit{D. Gaboriau} and \textit{F. Paulin} [Geom. Dedicata 88, No. 1-3, 153-197 (2001; Zbl 1020.51007)]. The content of this paper is to establish an analogous result for the three-dimensional case. In order to obtain it the authors combine a classification theorem for Coxeter polytopes of spherical type and an arborescent construction of buildings of splittable Coxeter systems.
locally finite buildings of spherical type
classification theorem for Coxeter polytopes of spherical type
scientific article; zbMATH DE number 1889248