On axiomatic definitions of non-discrete affine buildings

TL;DR

This paper establishes the equivalence of various axiomatic definitions of non-discrete affine buildings, demonstrating that the Euclidean building definition depends solely on the topological class of the model space's metric.

Contribution

It provides new proofs of axiomatic equivalence for affine buildings and clarifies the metric conditions necessary for a space to qualify as a building.

Findings

Axioms for non-discrete affine buildings are equivalent under different metric and exchange conditions.

The Euclidean building definition depends only on the topological equivalence class of the model space's metric.

A space can only be a building if the induced metric satisfies the triangle inequality.

Abstract

In this paper we prove equivalence of sets of axioms for non-discrete affine buildings, by providing different types of metric, exchange and atlas conditions. We apply our result to show that the definition of a Euclidean building depends only on the topological equivalence class of the metric on the model space. The sharpness of the axioms dealing with metric conditions is illustrated in an appendix. There it is shown that a space X defined over a model space with metric d is possibly a building only if the induced distance function on X satisfies the triangle inequality.