Compact totally disconnected Moufang buildings. (English) Zbl 1269.20024
By the work of J. Tits, [Buildings of spherical types and finite \(BN\)-pairs. Lect. Notes Math. 386. Springer-Verlag (1974; Zbl 0295.20047)], it is possible to classify all spherical buildings of rank \(>3\), and all spherical buildings of rank \(2\) with the Moufang condition. However, many interesting examples of spherical buildings appear with an additional feature: they are topological buildings. For example, the boundary at infinity of a symmetric space, or a Euclidean building, is a topological spherical building. Furthermore, assuming the space is locally compact, this building will be compact. Therefore, one can wonder which spherical buildings admit a compact topology. Several authors worked on this problem in special cases, mostly when the building is connected or locally connected: see for example K. Burns and R. Spatzier [Publ. Math., Inst. Hautes Étud. Sci. 65, 5-34 (1987; Zbl 0643.53036)], and T. Grundhöfer, N. Knarr and L. Kramer [Geom. Dedicata 83, No. 1-3, 1-29 (2000; Zbl 0974.51014)].
The paper under review extends these results to the general case. The authors prove that a compact infinite irreducible spherical building \(\Delta\) which is Moufang is always the boundary of a symmetric space or of a Bruhat-Tits building. Furthermore, the topology of \(\Delta\) is unique (meaning that every automorphism is continuous), except maybe for the conjugation in complex symmetric spaces.
The paper under review extends these results to the general case. The authors prove that a compact infinite irreducible spherical building \(\Delta\) which is Moufang is always the boundary of a symmetric space or of a Bruhat-Tits building. Furthermore, the topology of \(\Delta\) is unique (meaning that every automorphism is continuous), except maybe for the conjugation in complex symmetric spaces.
Reviewer: Jean Lécureux (Orsay)
MSC:
20E42 | Groups with a \(BN\)-pair; buildings |
51E24 | Buildings and the geometry of diagrams |
22F50 | Groups as automorphisms of other structures |
51H20 | Topological geometries on manifolds |
53C35 | Differential geometry of symmetric spaces |
Keywords:
Moufang buildings; spherical buildings; topological buildings; compact buildings; locally compact groups; symmetric spacesReferences:
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