Generalized polygons with non-discrete valuation defined by two-dimensional affine R-buildings

Abstract

In this paper, we show that the building at infinity of a two-dimensional affine R-building is a generalized polygon endowed with a valuation satisfying some specific axioms. Specializing to the discrete case of affine buildings, this solves part of a long standing conjecture about affine buildings of type (G) over tilde (2), and it reproves the results obtained mainly by the second author for types (A) over tilde (2) and (C) over tilde (2). The techniques are completely different from the ones employed in the discrete case, but they are considerably shorter, and general (i.e., independent of the type of the two-dimensional R-building)