Automorphisms and opposition in twin buildings (Q2861574)

Motivated by recent results on the size of the set of chambers mapped to an opposite chamber by an automorphism of certain projective and polar spaces and generalized quadrangles (see, for example, [\textit{B. Temmermans} et al., Ann. Comb. 16, No. 4, 905--916 (2012; Zbl 1259.51004)]) the authors ask the question `How many chambers or, more generally, residues can be mapped to opposite ones?'. Their answers to this question are threefold. (1) Every automorphism of a thick twin building that interchanges the two halves of the building maps some spherical residue to an opposite residue. (2) An automorphism \(\theta\) of a twin building maps all residues of fixed type to opposite residues if and only if it maps all chambers to opposite chambers. Moreover, if \(\theta\) maps all chambers to opposites, then \(\theta\) is necessarily type-preserving, and maps all residues to opposite residues. (3) An automorphism of an irreducible 2-spherical locally finite thick twin building of rank at least 3 cannot map every residue of fixed type to an opposite residue. NEWLINENEWLINENEWLINEThe proof of (3) is reduced to the case of a twin building with at least one nonexotic rank-two residue which is not a generalised digon and uses the characterization obtained in (2). The authors furthermore present two applications of (3). An involution of a spherical building either maps every chamber to an opposite or fixes at least one simplex. In particular, every involution of a finite irreducible thick spherical building of rank at least 3 fixes some simplex. Secondly, an irreducible 2-spherical locally finite Kac-Moody group of rank at least 3 does not have a generalised Iwasawa decomposition.