we have ${\rm Aut}(\mathcal{B}) = \{1 \}$ (take for example a tree in which any two distinct vertices have distinct valencies). However, J. Tits’ classification of Euclidean buildings [@TitsCome] implies that in dimension $\geqslant 3$ any irreducible building comes – via Bruhat-Tits theory, see next remark – from a simple algebraic group over a local field, and therefore admits a large automorphism group. At last, note that there do exist 2-dimensional exotic Euclidean buildings, with interesting but unexpectedly small automorphism groups [@Barre].

\[rk - BrT\] In Sect. \[s - Bruhat-Tits general\], we will briefly introduce Bruhat-Tits theory. The main outcome of this important part of algebraic group theory is that, given a semisimple algebraic group $ {\rm G}$ over a local field $k$, there exists a discrete Euclidean building $\mathcal{B} = \mathcal{B}({\rm G},k)$ on which the group of rational points ${\rm G}(k)$ acts by isometries and [strongly transitively]{} (i.e., transitively on the inclusions of an alcove in an apartment).

Let ${\rm G}$ as above be the group ${\rm SL}_3$. Then the Euclidean building associated to ${\rm SL}_3$ is a Euclidean building in which every apartment is a Coxeter complex of type $\widetilde{{\rm A}_2}$, that is the previously described $2$-dimensional tiling 