On the congruence kernel for simple algebraic groups

This paper contains several results about the structure of the congruence kernel $C^{(S)}(G)$ of an absolutely almost simple simply connected algebraic group $G$ over a global field $K$ with respect to a set of places $S$ of $K$. In particular, we show that $C^{(S)}(G)$ is always trivial if $S$ contains a generalized arithmetic progression. We also give a criterion for the centrality of $C^{(S)}(G)$ in the general situation in terms of the existence of commuting lifts of the groups $G(K_v)$ for $v\notin S$ in the $S$-arithmetic completion $\widehat {G}^{(S)}$. This result enables one to give simple proofs of the centrality in a number of cases. Finally, we show that if $K$ is a number field and $G$ is $K$-isotropic, then $C^{(S)}(G)$ as a normal subgroup of $\widehat {G}^{(S)}$ is almost generated by a single element.