An approximation principle for congruence subgroups

The motivating question of this paper is roughly the following: given a flat group scheme G over Z(p), p prime, with semisimple generic fiber G(Qp), how far are open subgroups of G(Z(p)) from subgroups of the form X(Z(p))K-p(p(n)), where X is a subgroup scheme of G and K-p(p(n)) is the principal congruence subgroup Ker(G(Z(p)) -> G(Z/p(n)Z))? More precisely, we will show that for G(Qp) simply connected there exist constants J >= 1 and epsilon > 0, depending only on G, such that any open subgroup of G(Z(p)) of level p(n) admits an open subgroup of index

We also give a correspondence between natural classes of Z(p)-Lie subalgebras of gZ(p) and of closed subgroups of G(Z(p)) that can be regarded as a variant over Z(p) of Nori's results on the structure of finite subgroups of GL(N-0,F-p) for large p [Nor87].

As an application we give a bound for the volume of the intersection of a conjugacy class in the group G((Z) over cap = Pi(p) G(Z(p)), for G defined over Z, 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 G(Z).