Moufang buildings and twin buildings (Q5948412)

Twin buildings are generalizations of spherical buildings and have been introduced by M. Ronan and J. Tits in order to study Kac-Moody groups from a geometrical point of view. In the paper under review the author deals with the question of whether every Moufang building corresponds to a twin building. In [Lond. Math. Soc. Lect. Note Ser. 165, 249-286 (1992; Zbl 0851.22023)] \textit{J. Tits} stated that this is actually the case and refers to his earlier article [J. Algebra 105, 542-573 (1987; Zbl 0626.22013)] which indeed contains a rough outline for a possible proof. In the paper under review the author provides a rigorous proof by following this outline. However, she uses alternative geometric arguments which give a better insight into the geometry of twin buildings and their relation with groups. The author constructs a chamber system \({\mathcal C}^-\) using the root groups of the Moufang building \((\Delta,W,S,d)\). The chambers are certain right cosets in the group \(U_-\) generated by all root groups to negative roots. The map \(\kappa\) from \({\mathcal C}^-\) to \(\Delta\) which sends a coset associated with \(w\in W\) to its image of the chamber \(w(c_+)\), where \(c_+\) is the standard chamber in \(\Delta\), becomes a 2-covering from \({\mathcal C}^-\) to \(\Delta\) and thus is an isomorphism. From the Moufang building \(\Delta\) one therefore obtains two \(BN\)-pairs \((G,B_\pm,N,S)\) with associated buildings \((\Delta_\pm, W,S,d_\pm)\). The author defines a symmetric relation \({\mathcal O}\) on chambers of the buildings \(\Delta_\pm\) that contains exactly those pairs that can be mapped under \(G\) to the standard pairs \((c_\pm,c_\mp)\) and shows that \({\mathcal O}\) is a twinning between \(\Delta_+\) and \(\Delta_-\) in case of a rank 2 Moufang building of spherical type and an opposition relation of a twinning between \(\Delta_+\) and \(\Delta_-\) in case of a non-spherical rank 2 Moufang building. Using a characterisation of the opposition relation by \textit{B. Mühlherr} [Eur. J. Comb. 19, 603-612 (1998; Zbl 0915.51006)] it follows that \({\mathcal O}\) defines a twinning between \(\Delta_+\) and \(\Delta_-\) and hence \(\Delta\cong\Delta_+\) is half of a twin building.