# An approximation principle for congruence subgroups

### Tobias Finis

Universität Leipzig, Germany### Erez Lapid

Weizmann Institute of Science, Rehovot, Israel

## Abstract

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 $G_{Q_{p}}$, 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 $G_{Q_{p}}$ simply connected there exist constants $J≥1$ and $ε>0$, 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 $X(Z_{p})K_{p}(p_{⌈εn⌉})$ for some proper, connected algebraic subgroup $X$ of $G$ defined over $Q_{p}$. Moreover, if $G$ is defined over $Z$, then $ε$ and $J$ can be taken independently of $p$.

We also give a correspondence between natural classes of $Z_{p}$-Lie subalgebras of $g_{Z_{p}}$ 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 $p$ [Nor 87].

As an application we give a bound for the volume of the intersection of a conjugacy class in the group $G(Z^)=∏_{p}G(Z_{p})$, 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 $G(Z)$.

## Cite this article

Tobias Finis, Erez Lapid, An approximation principle for congruence subgroups. J. Eur. Math. Soc. 20 (2018), no. 5, pp. 1075–1138

DOI 10.4171/JEMS/783