Buildings, valuated matroids, and tropical linear spaces Battistella, Luca Kühn, Kevin Kuhrs, Arne Ulirsch, Martin Vargas, Alejandro <jats:title>Abstract</jats:title><jats:p>Affine Bruhat–Tits buildings are geometric spaces extracting the combinatorics of algebraic groups. The building of parameterizes flags of subspaces/lattices in or, equivalently, norms on a fixed finite‐dimensional vector space, up to homothety. It has first been studied by Goldman and Iwahori as a piecewise‐linear analogue of symmetric spaces. The space of seminorms compactifies the space of norms and admits a natural surjective restriction map from the Berkovich analytification of projective space that factors the natural tropicalization map. Inspired by Payne's result that the analytification is the limit of all tropicalizations, we show that the space of seminorms is the limit of all tropicalized <jats:italic>linear</jats:italic> embeddings and prove a faithful tropicalization result for compactified linear spaces. The space of seminorms is in fact the tropical linear space associated to the universal realizable valuated matroid.</jats:p> Wiley 2023 info:eu-repo/semantics/article doc-type:article text http://purl.org/coar/resource_type/c_6501 https://ris.uni-paderborn.de/record/66358 Battistella L, Kühn K, Kuhrs A, Ulirsch M, Vargas A. Buildings, valuated matroids, and tropical linear spaces. <i>Journal of the London Mathematical Society</i>. 2023;109(1). doi:<a href="https://doi.org/10.1112/jlms.12850">10.1112/jlms.12850</a> eng info:eu-repo/semantics/altIdentifier/doi/10.1112/jlms.12850 info:eu-repo/semantics/altIdentifier/issn/0024-6107 info:eu-repo/semantics/altIdentifier/issn/1469-7750 info:eu-repo/semantics/closedAccess