Solution of the congruence subgroup problem for solvable algebraic groups

Source abstract

Let k be an algebraic number field of finite degree over the field Q of rational numbers. We denote by o the ring of integers in k. In general, for a subring A, containing 1, of a universal domain Ω we denote by GL(n, A ) the subgroup of GL(n, Ω ) consisting of matrices x = ( x ij ) with x ij ∈ A and det x ∈ A × , the group of units of A. Now, we consider an algebraic group G in GL(n, Ω ) defined over k.