Two-Dimensional Ane R-Buildings De-ned by Generalized Polygons with Non-Discrete Valuation

Abstract

In this paper we complete the proof of the 'equivalence' of non-discrete R-buildings of types (A) over tilde (2) and (C) over tilde (2), with, respectively, projective planes and generalized quadrangles with non-discrete valuation, begun in [7]. We also complete the proof of the 'equivalence' of an affine building of rank 3 with a generalized polygon with discrete valuation (by proving this for generalized hexagons), begun in [14]. We also complement the main result of [13] by proving uniqueness up to scalar multiples of the weight sequences of polygons with non-discrete valuation. As an application, we produce some new explicitly defined non-discrete R-buildings, in particular a class of type (A) over tilde (2) with arbitrary residues.