of the Euclidean space ${\bf R}^2$ by regular triangles. Strong transitivity of the ${\rm SL}_3(k)$-action means here that given any alcoves (triangles) $c, c'$ and any apartments $\mathbb{A}, \mathbb{A}'$ such that $c \subset \mathbb{A}$ and $c' \subset \mathbb{A}'$ there exists $g \in {\rm SL}_3(k)$ such that $c'=g.c$ and $\mathbb{A}'=g.\mathbb{A}$.

The description of the apartments doesn’t depend on the local field $k$ (only on the Dynkin diagram of the semisimple group in general), but the field $k$ plays a role when one describes the combinatorial neighborhoods of facets, or small metric balls around vertices. Such subsets, which intersect finitely many facets when $k$ is a local field, are known to be realizations of some (spherical) buildings: these buildings are naturally associated to semisimple groups (characterized by some subdiagram of the Dynkin diagram of ${\rm G}$) over the residue field $\widetilde{k}$ of $k$.

\[ex - SL3bis\] For ${\rm G}={\rm SL}_3$ and $k={\bf Q}_p$, each sufficiently small ball around a vertex is the flag complex of a $2$-dimensional vector space over ${\bf Z}/p{\bf Z}$ and any edge in the associated Bruhat-Tits building is contained in the closure of exactly $p+1$ triangles. A suitably small metric ball around any point in the relative interior 