On axiomatic definitions of non-discrete affine buildings

Details On axiomatic definitions of non-discrete affine buildings On axiomatic definitions of non-discrete affine buildings

2010-03-16

arxiv.org/abs/1003.3050v2

Mathematics

Metric Geometry

Errors corrected, resultsextended. (This replaces the two earlier, separate preprints "Axioms of affine buidlings" arXiv:0909.

Scientific paper

In this paper we provide six different equivalent sets of axioms for affine $\Lambda$-buildings, by providing different types of metric conditions, exchange conditions, and atlas conditions. The work on atlas conditions builds on the work of Anne Parreau on equivalence of axioms for Euclidean buildings. The new axiom systems provide for potentially easier proofs that spaces are $\Lambda$-buildings. Moreover, 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 of the building. In an appendix a class of examples is constructed to illustrate the sharpnes of the axioms dealing with metric conditions. These examples show that a space $X$ defined over a given model space (with metric $d$) is possibly a building only if the distance function induced on $X$ (by $d$) satisfies the triangle inequality.

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