  Given a Chevalley group $\mathcal{G}$ of classical type and a Borel subgroup
$\mathcal{B} \subseteq \mathcal{G}$, we compute the $\Sigma$-invariants of the
$S$-arithmetic groups $\mathcal{B}(\mathbb{Z}[1/N])$, where $N$ is a product of
large enough primes. To this end, we let $\mathcal{B}(\mathbb{Z}[1/N])$ act on
a Euclidean building $X$ that is given by the product of Bruhat--Tits buildings
$X_p$ associated to $\mathcal{G}$, where $p$ runs over the primes dividing $N$.
In the course of the proof we introduce necessary and sufficient conditions for
convex functions on $\mbox{CAT(0)}$-spaces to be continuous. We apply these
conditions to associate to each simplex at infinity $\tau \subset
\partial_{\infty} X$ its so-called parabolic building $X^{\tau}$, which we
study from a geometric point of view. Moreover, we introduce new techniques in
combinatorial Morse theory, which enable us to take advantage of the concept of
essential $n$-connectivity rather than actual $n$-connectivity. Most of our
building theoretic results are proven in the general framework of spherical and
Euclidean buildings. For example, we prove that the complex opposite each
chamber in a spherical building $\Delta$ contains an apartment, provided
$\Delta$ is thick enough and $\mbox{Aut}(\Delta)$ acts chamber transitively on
$\Delta$.
