The congruence subgroup problem and the fundamental group of the reductive Borel-Serre compactification

Details The congruence subgroup problem and the fundamental group of the reductive Borel-Serre compactification The congruence subgroup problem and the fundamental group of the reductive Borel-Serre compactification

2011-06-23

arxiv.org/abs/1106.4810v1

Mathematics

Geometric Topology

25 pages, 1 figure, uses Xy-pic 3.8.6

Scientific paper

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 to identify the congruence subgroup kernel with the fundamental group of the reductive Borel-Serre compactification of \Gamma \backslash X for certain sufficiently small S-arithmetic subgroups \Gamma of G. Our result follows from an explicit computation of the fundamental group of the reductive Borel-Serre compactification of \Gamma \backslash X. 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. Similar computations of the fundamental group of the Satake compactifications are made.

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