Math @ Duke

Publications [#213234] of Leslie Saper
Preprints
 L. Ji, K. Murty, L. Saper, and J. Scherk, The Congruence Subgroup Kernel and the Fundamental Group of the Reductive BorelSerre Compactification
(June, 2011) [arXiv:1106.4810]
(last updated on 2013/03/22)
Author's Comments: We have revised and renamed the paper to clarify its focus on calculating the fundamental group. The new version does not have the expository material on the congruence subgroup problem so this version may still be of interest.
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 ∈ S of the symmetric spaces
associated to G(k_{v}), when v is an infinite place, and the BruhatTits
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 BorelSerre compactification of Γ \ X for
certain sufficiently small Sarithmetic subgroups Γ of G. Our
result follows from explicit computations of the fundamental group of the
reductive BorelSerre compactifications of Γ \ X. In the case that Γ is neat, we show that this fundamental group is isomorphic to Γ / EΓ, where EΓ is the subgroup generated by the elements of Γ belonging to unipotent radicals of parabolic ksubgroups. Similar
computations of the fundamental group of the Satake
compactifications are made.


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