Given a Chevalley group
G
\mathcal {G}
of classical type and a Borel subgroup
B
⊆
G
\mathcal {B} \subseteq \mathcal {G}
, we compute the
Σ
\Sigma
-invariants of the
S
S
-arithmetic groups
B
(
Z
[
1
/
N
]
)
\mathcal {B}(\mathbb {Z}[1/N])
, where
N
N
is a product of large enough primes. To this end, we let
B
(
Z
[
1
/
N
]
)
\mathcal {B}(\mathbb {Z}[1/N])
act on a Euclidean building
X
X
that is given by the product of Bruhat–Tits buildings
X
p
X_p
associated to
G
\mathcal {G}
, where
p
p
is a prime dividing
N
N
. In the course of the proof we introduce necessary and sufficient conditions for convex functions on
C
A
T
(
0
)
CAT(0)
-spaces to be continuous. We apply these conditions to associate to each simplex at infinity
τ
⊂
∂
∞
X
\tau \subset \partial _\infty X
its so-called parabolic building
X
τ
X^{\tau }
and to study it 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
n
-connectivity rather than actual
n
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
Δ
\Delta
contains an apartment, provided
Δ
\Delta
is thick enough and
A
u
t
(
Δ
)
Aut(\Delta )
acts chamber transitively on
Δ
\Delta
.