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 ∉ S in the S-arithmetic completion Ĝ (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 Ĝ (S) is almost generated by a single element.