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Publications [#213234] of Leslie Saper


  1. L. Ji, K. Murty, L. Saper, and J. Scherk, The Congruence Subgroup Kernel and the Fundamental Group of the Reductive Borel-Serre 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.

    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(kv), when v is an infinite place, and the Bruhat-Tits buildings associated to G(kv), 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 Borel-Serre compactification of Γ \ X for certain sufficiently small S-arithmetic subgroups Γ of G. Our result follows from explicit computations of the fundamental group of the reductive Borel-Serre 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 k-subgroups. Similar computations of the fundamental group of the Satake compactifications are made.
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