$W$ is $\overline W$ and such that there exists a point $x \in \Sigma$ and a subgroup ${\rm T} \subset {\rm Isom}(\Sigma)$ of translations satisfying $W = W_x \cdot {\rm T}$; we use here the notation $W_x = {\rm Stab}_W(x)$. A point $x$ satisfying this condition is called *special*.

\[defi - non-simplicial building\] Let $\mathcal{B}$ be a set and let $\mathcal{A} = \{f : \Sigma \to \mathcal{B} \}$ be a collection of injective maps, whose images are called [apartments]{}. We say that $\mathcal{B}$ is a [Euclidean building]{} of type $(\Sigma,W)$ if the apartments satisfy the following axioms.

-   The family $\mathcal{A}$ is stable by precomposition with any element of $W$ (i.e., for any $f \in \mathcal{A}$ and any $w \in W$, we have $f \circ w \in \mathcal{A}$).

-   For any $f,f' \in \mathcal{A}$ the subset $\mathcal{C}_{f,f'}= f'^{-1} \bigl( f(\Sigma) \bigr)$ is convex in $\Sigma$ and there exists $w \in W$ such that we have the equality of restrictions $(f^{-1} \circ f') \mid_{\mathcal{C}_{f,f'}} = w \mid_{\mathcal{C}_{f,f'}}$.

-   Any two points of $\mathcal{B}$ are contained in a suitable apartment.

At this stage, there is a well-defined map $d : \mathcal{B}\times \mathcal{B} \to {\bf R}_{\geqslant 0}$ and we further require:

-   Given 