Abstract

In this paper, we study an algebraically closed field \ expanded by two unary predicates denoting an algebraically closed proper subfield k and a multiplicative subgroup \. This will be a proper expansion of algebraically closed field with a group satisfying the Mann property, and also pairs of algebraically closed fields. We first characterize the independence in the triple \\). This enables us to characterize the interpretable groups when \ is divisible. Every interpretable group H in \\) is, up to isogeny, an extension of a direct sum of k-rational points of an algebraic group defined over k and an interpretable abelian group in \ by an interpretable group N, which is the quotient of an algebraic group by a subgroup \, which in turn is isogenous to a cartesian product of k-rational points of an algebraic group defined over k and an interpretable abelian group in \.