Abstract properties of $S$-arithmetic groups and the congruence s problem

Suppose $G$ is a simple and simply connected algebraic group over an algebraic number field $K$ and $S$ is a finite set of valuations of $K$ containing all Archimedean valuations. This paper is a study of the connections between abstract properties of the $S$-arithmetic subgroup $\mathbf\Gamma=G_{O(S)}$ and the congruence property, i.e. the finiteness of the corresponding congruence kernel $C=C^S(G)$. In particular, it is shown that if the profinite completion $\Delta=\widehat\Gamma$ satisfies condition $(\mathbf {PG})'$, (i.e., for any integer $n>0$ and any prime $p$ there exist $c$ and $k$ such that $|\Delta/\Delta^{np^\alpha}|\leqslant cp^{k\alpha}$ for all $\alpha>0$, then $C$ is finite. Examples are given demonstrating the possibility of effectively verifying $(\mathbf {PG})'$ .