diameter $\leq\pihalf$ iff $W$ fixes no point in $\amod$, equivalently, the Coxeter complex does not split off a spherical join factor (in the category of Coxeter complexes). In this case, the collection of chambers defines on $\amod$ the structure of a simplicial complex, the simplices being intersections of chambers.

Every chamber is a fundamental domain for the action $W\acts\amod$. The [*spherical model chamber*]{} can be defined as the quotient $\simod=\amod/W$. We identify it with a chamber in the model apartment, $\simod\subset\amod$, which we refer to as the [*fundamental chamber*]{}.

We call the natural projection $$\theta:\amod\to\amod/W\cong\simod$$ the [*type*]{} map for $\amod$. It restricts to an isometry on every chamber. A [*face type*]{} is a face of $\simod$. The type of a simplex $\bar\tau\subset\amod$ is then defined as $\theta(\bar\tau)$. Throughout the paper, we will use the notation $\taumod,\taumod',\numod,\numod',\ldots$ for face types. Furthermore, we will denote by $\Wt\leq W$ the stabilizer of the face type $\taumod\subseteq\simod$.

The [*longest element*]{} of the Weyl group is the unique element $w_0\in W$ sending $\simod$ to the opposite chamber $-\simod$. The [*standard involution*]{} (also known as the [*opposition involution*]{}) of the model chamber is given by $\iota:=-w_0:\simod\to\simod$.

### Spherical buildings {#sec:sphbld}

A [*spherical building*]{} modeled on a 