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# Asian Journal of Mathematics

## Volume 19 (2015)

### Number 3

### The fundamental group of reductive Borel–Serre and Satake compactifications

Pages: 465 – 486

DOI: http://dx.doi.org/10.4310/AJM.2015.v19.n3.a4

#### Authors

#### Abstract

Let $ \mathbf{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 $\mathbf{G}(k_v)$, when $v$ is an infinite place, and the Bruhat–Tits buildings associated to $\mathbf{G}(k_v)$, 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 $\Gamma \setminus X$, where $\Gamma$ is an $S$-arithmetic subgroup of $\mathbf{G}$. 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 $C(S, \mathbf{G})$ yield similar results.

#### Keywords

fundamental group, reductive Borel-Serre compactification, Bruhat-Tits buildings, congruence subgroup kernel

#### 2010 Mathematics Subject Classification

Primary 20F34, 22E40, 22F30. Secondary 14M27, 20G30.