TY - JOUR A1 - Battistella, Luca A1 - Kühn, Kevin A1 - Kuhrs, Arne A1 - Ulirsch, Martin A1 - Vargas De León, Alejandro José T1 - Buildings, valuated matroids, and tropical linear spaces T2 - Journal of the London Mathematical Society N2 - Affine Bruhat--Tits buildings are geometric spaces extracting the combinatorics of algebraic groups. The building of PGL parametrizes 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 linear embeddings ι:Pr↪Pn 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. Y1 - 2023 UR - http://publikationen.ub.uni-frankfurt.de/frontdoor/index/index/docId/82362 UR - https://nbn-resolving.org/urn:nbn:de:hebis:30:3-823628 SN - 1469-7750 SN - 0024-6107 VL - 109.2024 IS - e12850 PB - Wiley CY - Oxford ER -