An approximation principle for congruence subgroups

作者：Finis Tobias^*; Lapid Erez

来源：Journal of the European Mathematical Society, 2018, 20(5): 1075-1138.

DOI：10.4171/JEMS/783

• Lattices in Lie groups
• uniform pro-p groups
• Lie algebras

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 <= J which is contained in X(Z(p)) K-p (p (inverted right perpendicular) (epsilon n) (inverted left perpendicular)) for some proper, connected algebraic subgroup X of G defined over Q(p). Moreover, if G is defined over Z, then epsilon and J can be taken independently of p.
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).

• 出版日期2018