Two-dimensional affine \(\mathbb R\)-buildings defined by generalized polygons with non-discrete valuation (Q635873)

An important tool in the study of affine buildings is their building at infinity. In the rank \(3\) case, this is a ``generalized polygon'', but not all generalized polygons may appear as boundaries of buildings. It turns out that the additional structure need for a generalized polygon to be the boundary of some building is a ``discrete valuation''. This notion was introduced by the second author in [\textit{H. van Maldeghem}, Arch. Math. 53, No.~3, 513--520 (1989; Zbl 0659.51010)]. The result mentioned above was conjectured in [loc. cit.]; the paper under review completes its proof. This paper also treats with non-discrete affine buildings. In this case, the relevant notion is the one of ``generalized polygon with real valuation'', introduced by the author in [\textit{K. Struyve} and \textit{H. Van Maldeghem}, Adv. Geom. 10, No. 3, 465--476 (2010; Zbl 1207.51010)]. However, they are not able to prove in the general case that any generalized \(n\)-gon is the boundary of a non-discrete affine building: they give a proof only in the cases when \(n=3\) or \(4\).