The fundamental group of reductive Borel-Serre and Satake compactifications

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 $\mathrm{\Gamma }\mathrm{\backslash }X$, where $\mathrm{\Gamma }$ is an $S$-arithmetic subgroup of $G$. In the case that $\mathrm{\Gamma }$ is neat, we show that this fundamental group is isomorphic to $\mathrm{\Gamma }\mathrm{/}E\mathrm{\Gamma }$, where $E\mathrm{\Gamma }$ is the subgroup generated by the elements of $\mathrm{\Gamma }$ 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.