On the congruence kernel for simple algebraic groups

Abstract

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(Kv) 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.