  Let $G$ be an almost simple, simply connected algebraic group defined over a
number field $k$, and let $S$ be a finite set of places of $k$ including all
infinite places. Let $X$ be the product over $v\in S$ of the symmetric spaces
associated to $G(k_v)$, when $v$ is an infinite place, and the Bruhat-Tits
buildings associated to $G(k_v)$, when $v$ is a finite place. The main result
of this paper is an explicit computation of the fundamental group of the
reductive Borel-Serre compactification of $\Gamma\backslash X$, where $\Gamma$
is an $S$-arithmetic subgroup of $G$. In the case that $\Gamma$ is neat, we
show that this fundamental group is isomorphic to $\Gamma/E\Gamma$, where
$E\Gamma$ is the subgroup generated by the elements of $\Gamma$ belonging to
unipotent radicals of $k$-parabolic subgroups. Analogous computations of the
fundamental group of the Satake compactifications are made. It is noteworthy
that calculations of the congruence subgroup kernel $C(S,G)$ yield similar
results.
