On almost strong approximation for some exceptional groups

Abstract

If G is a simply connected semisimple group defined over a number field k and ∞ is the set of all infinite places of k , then G has strong approximation with respect to ∞ if and only if the archimedean part of any k -simple component of the adèle group G A is non-compact. Using the affine Bruhat–Tits building, the authors of [W.K. Chan, J. Hsia, On almost strong approximation of algebraic groups, J. Algebra 254 (2002) 441] formulated an almost strong approximation property (ASAP) for groups of compact type, and they proved that ASAP holds for all classical groups of compact type whose Tits indices over k are not 2 A n (d) with d ⩾3. In this paper, we show that ASAP holds for groups of types 3,6 D 4 ,G 2 ,F 4 ,E 7 , or E 8 .