Two-dimensional affine R-buildings defined by generalized polygons with non-discrete valuation

TL;DR

This paper completes the proof of the equivalence between certain non-discrete affine buildings and generalized polygons with non-discrete valuation, and introduces new explicitly defined non-discrete R-buildings with diverse residues.

Contribution

It finalizes the proof of the equivalence for types A~_2 and C~_2, and extends the classification of non-discrete R-buildings, including new examples with arbitrary residues.

Findings

Proved the equivalence of non-discrete R-buildings of types A~_2 and C~_2 with projective planes and generalized quadrangles.

Extended the equivalence to affine buildings of rank 3 with generalized hexagons.

Constructed new non-discrete R-buildings with arbitrary residues.

Abstract

In this paper we complete the proof of the "equivalence" of non-discrete R-buildings of types A~_2 and C~_2, with, respectively, projective planes and generalized quadrangles with non-discrete valuation, begun in previous paper of the authors. 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). We also complement a result of the second author 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~_2 with arbitrary residues.