singular spaces but, by and large, play a similar role for non-Archimedean Lie groups ${\rm G}(k)$ as above.

### Simplicial definition {#sss - simplicial}

The general reference for building theory from the various “discrete” viewpoints is [@AbramenkoBrown]. Let us start with an affine reflection group, more precisely a [*Coxeter group of affine type*]{} [@Lie456]. The starting point to introduce this notion is a locally finite family of hyperplanes – called [*walls*]{} – in a Euclidean space \[[**loc. cit.**]{}, V §1 introduction\]. An affine Coxeter group can be seen as a group generated by the reflections in the walls, acting properly on the space and stabilizing the collection of walls \[[**loc. cit.**]{}, V §3 introduction\]; it is further required that the action on each irreducible factor of the ambient space be via an infinite [*essential*]{} group (no non-zero vector is fixed by the group).

\[ex - simplicial apartments\]

-   The simplest (one-dimensional) example of a Euclidean tiling is provided by the real line tesselated by the integers. The corresponding affine Coxeter group, generated by the reflections in two consecutive vertices (i.e., integers), is the infinite dihedral group ${\rm D}_\infty$.

-   The next simplest (irreducible) example is provided by the tesselation of 