Asian Journal of Mathematics

Volume 19 (2015)

Number 3

The fundamental group of reductive Borel–Serre and Satake compactifications

Pages: 465 – 486

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

Authors

Lizhen Ji (Department of Mathematics, University of Michigan, Ann Arbor, Mich., U.S.A.)

V. Kumar Murty (Department of Mathematics, University of Toronto,, Ontario, Canada)

Leslie Saper (Department of Mathematics, Duke University, Durham, North Carolina, U.S.A.)

John Scherk (Department of Computer and Mathematical Sciences, University of Toronto Scarborough, Toronto, Ontario, Canada)

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.

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Published 19 June 2015