The fundamental group of reductive Borel-Serre and Satake compactifications
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 $v\in S$ 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.
- Publication:
-
arXiv e-prints
- Pub Date:
- June 2011
- DOI:
- 10.48550/arXiv.1106.4810
- arXiv:
- arXiv:1106.4810
- Bibcode:
- 2011arXiv1106.4810J
- Keywords:
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- Mathematics - Geometric Topology;
- Mathematics - Group Theory;
- Mathematics - Number Theory;
- 20F34;
- 22E40;
- 22F30 (Primary);
- 14M27;
- 20G30 (Secondary)
- E-Print:
- 21 pages, 1 figure, uses Xy-pic 3.8.6