particular, we show how to interpret suitable spaces of norms to obtain concrete examples of buildings in the case when the algebraic group under consideration is the special linear group of a vector space. These spaces of norms will naturally be extended to spaces of (homothety classes of) seminorms when buildings are considered in the context of analytic projective spaces.

Euclidean buildings {#ss - Euclidean buildings}
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Euclidean buildings are non-Archimedean analogues of Riemannian symmetric spaces of the non-compact type, at least in the following sense: if ${\rm G}$ is a simple algebraic group over a valued field $k$, Bruhat-Tits theory (often) associates to ${\rm G}$ and $k$ a metric space, called a Euclidean building, on which ${\rm G}(k)$ acts by isometries in a “very transitive” way. This is a situation which is very close to the one where a (non-compact) simple real Lie group acts on its associated (non-positively curved) Riemannian symmetric space. In this more classical case, the transitivity of the action, the explicit description of fundamental domains for specific (e.g., maximal compact) subgroups and some non-positive curvature arguments lead to deep conjugacy and structure results – see [@MaubonGrenoble] and [@ParadanGrenoble] for a modern account. Euclidean buildings are 