Characterisations of generalized polygons and opposition in rank 2 twin buildings (Q1865866)

In [Ann. Comb. 4, 125-137 (2000; Zbl 0961.51011)] and [Math. Z. 238, 187-203 (2001; Zbl 1004.51016)], the authors investigated the opposition relation and 1-twinnings in spherical and twin buildings and obtained a new characterization of twin buildings. In the paper under review they continue their analysis and develop further the general notion in the rank 2 situation. The first sections are written in the language of (rank 2) incidence geometry and develop characterizations of the opposition relation in generalized polygons and of the generalized polygons themselves among rank 2 geometries in terms of the opposition relation and 1-twinnings. This eventually results in a considerable weakening of the classical definition of generalized polygons: If \(\Gamma\) is a rank 2 geometry with finite diameter \(n\) such that for every pair of elements \(x,y\) of \(\Gamma\), with \(x\) and \(y\) at distance \(n-1\) from each other, there is a unique element \(x'\) incident with \(x\) and nearest to \(y\) (in the incidence graph), then \(\Gamma\) is a generalized \(n\)-gon. The authors further present counter-examples that show that a 1-twinning of a rank 2 building need not be a twinning thus answering in the negative a question by \textit{B. Mühlherr} [Eur. J. Comb. 19, 603-612 (1998; Zbl 0915.51006)] whether twin buildings can be characterized by just using the 1-twinning property. In the final section the authors extend their considerations to spherical buildings of higher rank to obtain a characterization of the natural opposition relations of thick spherical buildings (up to equivalence) as those relations on the set of chambers which by doubling the building become the opposition relation of a twin building.