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

TL;DR

This paper demonstrates that the building at infinity of a two-dimensional affine R-building forms a generalized polygon with a valuation, providing new insights into affine buildings and solving longstanding conjectures.

Contribution

It introduces a novel approach to understanding the structure of affine R-buildings and generalizes previous results to a broader, type-independent context.

Findings

The building at infinity is a generalized polygon with a valuation.

It solves part of a longstanding conjecture for affine buildings of type G~_2.

It reproves known results for types A~_2 and C~_2 using new techniques.

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~_2, and it reproves the results obtained mainly by the second author for types A~_2 and C~_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).