Subgroups of algebraic groups which are clopen in the S-congruence topology

Abstract

Abstract. Let K be a global field and S be a finite set of places of K which includes all those of archimedean type. Let G be an algebraic group over K and G K be its K-rational points. The authors provide a detailed proof of a lemma of Raghunathan which states that (under fairly weak restrictions) the closure in the S-congruence topology of a subgroup of G K normalized by an S-arithmetic subgroup is also open. This leads to a significant simplification in the proof of one of the principal results in a recent joint paper of the authors.By applying the lemma to S-arithmetic lattices in G of K-rank one, where charðKÞ 0 0 and jSj ¼ 1, we can provide a lower estimate for the number of subgroups of a given index in such a lattice which are not S-congruence. This extends previous results of the first author and Andreas Schweizer.