The motivating question of this paper is roughly the following: given a flat group scheme over , prime, with semisimple generic fiber , how far are open subgroups of from subgroups of the form , where is a subgroup scheme of and is the principal congruence subgroup Ker? More precisely, we will show that for simply connected there exist constants and , depending only on , such that any open subgroup of of level admits an open subgroup of index which is contained in for some proper, connected algebraic subgroup of defined over . Moreover, if is defined over , then and can be taken independently of .
We also give a correspondence between natural classes of -Lie subalgebras of and of closed subgroups of that can be regarded as a variant over of Nori's results on the structure of finite subgroups of GL for large [Nor 87].
As an application we give a bound for the volume of the intersection of a conjugacy class in the group , for defined over , with an arbitrary open subgroup. In a companion paper, we apply this result to the limit multiplicity problem for arbitrary congruence subgroups of the arithmetic lattice .
Cite this article
Tobias Finis, Erez Lapid, An approximation principle for congruence subgroups. J. Eur. Math. Soc. 20 (2018), no. 5, pp. 1075–1138DOI 10.4171/JEMS/783