On axiomatic definitions of non-discrete affine buildings

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.Comment: Errors corrected, results extended. (This replaces the two earlier, separate preprints "Axioms of affine buidlings" arXiv:0909.2967v1 and "Affine $\Lambda$ buildings II" arXiv:0909.2059v1.