chamber $\si\subset \B$. We call the inverse $\kappa_{\si}=(\theta|_{\si})^{-1}: \si_{mod}\to\si$ the [*chart*]{} of the chamber $\si$. For a simplex $\tau\subset \B$, we call the face $\theta(\tau)\subseteq\simod$ the [*type*]{} of the simplex and $\kappa_{\tau}=(\theta|_{\tau})^{-1}:\theta(\tau)\to\tau$ its [*chart*]{}. We define the [*type*]{} of a point $\xi\in \B$ as its image $\theta(\xi)\in\simod$. A point $\xi\in \B$ is called [*regular*]{} if its type is an interior point of $\simod$, $\xi\in\inte(\simod)$, and [*singular*]{} otherwise.

We will sometimes say that a singular sphere has type $\taumod$ if it contains a top-dimensional simplex of type $\taumod$. (A singular sphere has in general several types.)

For a singular sphere $s\subset \B$, we define $\B(s)\subset \B$ as the union of all apartments containing $s$. It is a convex subset and splits off $s$ as a spherical join factor. Moreover, $\B(s)$ is a subbuilding, i.e. it inherits from $\B$ a spherical building structure modeled on the same Coxeter complex; the apartments of $\B(s)$ are precisely the apartments of $\B$ containing $s$. This building structure is however not thick, except in degenerate cases. In order to pass to a thick spherical building structure, take a maximal atlas of charts $\kappa:\amod\to \B(s)$ for which the maps $\kappa^{-1}|_s:s\to\amod$ coincide, and reduce the Weyl group to 