a model for an apartment can be taken to be a maximal flat in the symmetric space of ${\rm SL}_3({\bf R})/{\rm SO}(3)$ acted upon by its stabilizer in ${\rm SL}_3({\bf R})$ (using the notion of singular geodesics to distinguish the walls). There is a geometric way to define the Weyl group and Weyl chambers (six directions of simplicial cones) in this differential geometric context – see [@MaubonGrenoble] for the general case of arbitrary symmetric spaces.

Here is a (purely metric) definition of real trees. It is a metric space $({\rm X},d)$ with the following two properties:

-   it is [*geodesic*]{}: given any two points $x, x' \in {\rm X}$ there is a (continuous) map $\gamma : [0;d] \to {\rm X}$, where $d = d(x,x')$, such that $\gamma(0)=x$, $\gamma(d)=x'$ and $d\bigl(\gamma(s),\gamma(t) \bigr) = \, \mid\! s-t \!\mid$ for any $s,t \in [0;d]$;

-   any geodesic triangle is a tripod (i.e., the union of three geodesic segments with a common end-point).

\[rk - asymptotic cones\] Non-simplicial Euclidean buildings became more popular since recent work of geometric (rather than algebraic) nature, where non-discrete buildings appear as asymptotic cones of symmetric spaces and Bruhat-Tits buildings [@KleLee].

The remark implies in particular that there exist 