On almost strong approximation in reductive algebraic groups

A criterion for strong approximation in algebraic groups was obtained by V. P. Platonov in characteristic zero, and by G. A. Margulis and G. Prasad in positive characteristic. It follows from this criterion that strong approximation never holds for nonsimply connected groups (in particular, algebraic tori) and a finite set of places. We will report of a recent work where we show that a slightly weaker property, which we termed “almost strong approximation” can hold for nonsimply connected reductive groups and some special infinite sets of places. Applying this fact to maximal tori of an absolutely almost simple simply connected group, we generalize some results on the congruence subgroup problem. Joint work with Wojciech Tralle.