Presqu'un immeuble pour le groupe des automorphismes mod\'er\'es

Inspired by the Bruhat-Tits building of SL${}_n$(${\mathbb{Q}}_p$), we construct a complete metric space X with an action of the tame automorphism group of the affine space Tame($K^n$). The points in X are certain monomial valuations, and X admits a natural structure of Euclidean CW-complex of dimension n-1. When n = 3, and for K of characteristic zero, we prove that X has non-positive curvature and is simply connected, hence is a CAT(0) space. As an application we obtain the linearizability of finite subgroups in Tame($K^3$).