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Affine buildings for dihedral groups. (English) Zbl 1244.51003

By the foundational work of Tits [J. Tits, Buildings of Spherical Type and Finite BN-pairs. Lecture Notes in Mathematics. 386. Berlin-Heidelberg-New York: Springer-Verlag (1974; Zbl 0295.20047)], we know that thick spherical buildings in dimension \(\geq 2\) are all of algebraic origin, and hence can only have a crystallographic Weyl group. But, in dimension \(1\) (when the Coxeter group is a dihedral group), he provided a construction of thick building with arbitrary dihedral group.
One interest of spherical buildings is that some of them appear as boundaries of Euclidean buildings. For algebraic buildings, we know which spherical buildings can appear in this construction (namely, they are associated to an algebraic structure with an appropriate notion of a valuation); but this is not the case for non-algebraic spherical buildings.
There are some previous constructions of “exotic” Euclidean buildings not coming from algebraic groups, but so far all of them with crystallographic Weyl groups. The main result of the paper under review is a construction of non-discrete Euclidean buildings with an arbitrary dihedral Weyl group. The techniques rely mainly on a combinatorial convexity theorem for 2-dimensional CAT(0) spaces modeled on an affine Weyl group, together with a previous method of J. Tits [Invent. Math. 43, 283–295 (1977; Zbl 0399.20037)].

MSC:

51E24 Buildings and the geometry of diagrams
20E42 Groups with a \(BN\)-pair; buildings
53C20 Global Riemannian geometry, including pinching
20G15 Linear algebraic groups over arbitrary fields
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