The center conjecture for non-exceptional buildings
The centre conjecture for buildings states that a convex subcomplex \(\tilde\Delta\subset\Delta\) of a spherical building \({\pmb\Delta}=(\Delta,\subset)\) is completely reducible, that is, for each simplex \(A\in \tilde\Delta\) there is a simplex in \(\tilde\Delta\) which is opposite to \(A\) in \({\pmb\Delta}\), or is in the centre, that is, the stabilizer of \(\tilde\Delta\) in the automorphism group of \({\pmb\Delta}\) fixes a non-trivial simplex of \(\tilde\Delta\). The main result of the paper under review is that the centre conjecture holds for irreducible spherical buildings which are not of type \(E_6\), \(E_7\), \(E_8\) or \(F_4\). This then carries over to spherical buildings whose types have no direct factors \(E_6\), \(E_7\), \(E_8\) or \(F_4\). The proof of the main result follows ideas developed in [\textit{J.-P. Serre}, Astérisque 299, 195--217 (2005; Zbl 1156.20313)] and \textit{J. Tits} [Résumé de cours, in: Annuaire du Collège de France, 97e année, 89--102 (1996-97)]. It relies on the fact that buildings of finite rank are flag complexes of certain classes of incidence geometries, that is, generalized polygons, projective spaces and polar spaces, for which the authors derive combinatorial properties and give characterizations of opposite vertices in these buildings and of convex subcomplexes by closedness. Using an associated building of type \(F_4\) a proof of the centre conjecture for buildings of type \(D_4\) is outlined since this special case requires additional arguments as trialities may exist so that the automorphism group of such a building may be bigger.
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