The fundamental group of reductive Borel-Serre and Satake compactifications. (Q2515526)

Let \(k\) be a number field and let \(G\) be a connected, absolutely almost simple, simply connected algebraic group over \(k\). Let \(S\) be a finite set of places containing the set \(S_\infty\) of infinite places of \(k\). Assume that \(G\) is \(k\)-isotropic and has \(S\)-rank at least \(2\). One considers the product \(X:=X_\infty\times\prod_{v\in S\setminus S_\infty}X_v\) of the symmetric space \(X_\infty\) corresponding to \(G(k_\infty)\) and the Bruhat-Tits buildings \(X_v\) of \(G(k_v)\). Extending the work of Borel \& Serre and of Zucker, the authors construct the reductive Borel-Serre bordification \(\overline X^{RBS}\) of \(X\). For any \(S\)-arithmetic subgroup \(\Gamma\) of \(G(k)\), the action on \(X\) extends to one on \(\overline X^{RBS}\), and the quotient \(\Gamma\backslash\overline X^{RBS}\) is a compact, Hausdorff space, known as the Borel-Serre compactification of \(\Gamma\backslash X\). The main result of the paper is a computation of its fundamental group. When \(\Gamma\) is net, they show that the fundamental group is isomorphic to \(\Gamma/E\Gamma\) where \(E\Gamma\) is the subgroup of \(\Gamma\) generated by its elements which are in unipotent radicals of parabolic \(k\)-subgroups of \(G\). Further, the authors generalize the notion of Satake compactification to \(\Gamma\backslash X\), and compute their fundamental groups too; they turn out to be quotients of \(\Gamma/E\Gamma\). The authors observe the intriguing similarity between the above computations and the computation of the \(S\)-congruence kernel.