The fundamental group of reductive Borel-Serre and Satake compactifications

TL;DR

This paper explicitly computes the fundamental groups of reductive Borel-Serre and Satake compactifications for certain arithmetic quotients, revealing their structure in terms of arithmetic subgroups and unipotent elements.

Contribution

It provides the first explicit calculation of the fundamental group of these compactifications, linking it to arithmetic subgroup quotients and unipotent radicals.

Findings

Fundamental group of reductive Borel-Serre compactification is isomorphic to mma/Emma.

Fundamental group of Satake compactifications is similarly computed.

Results relate to the congruence subgroup kernel C(S,G).

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.