The fundamental group of reductive Borel–Serre and Satake compactifications
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Abstract
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 S of the symmetric spaces associated to G(kv ), when v is an infinite place, and the Bruhat-Tits buildings associated to G(kv), when v is a finite place. The main result of this paper is to compute explicitly the fundamental group of the reductive Borel-Serre compactification of \X, where is an S-arithmetic subgroup of G. In the case that is neat, we show that this fundamental group is isomorphic to /E, where E is the subgroup generated by the elements of 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.
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