Interpretable groups in Mann pairs

In this paper, we study an algebraically closed field \(\Omega \) expanded by two unary predicates denoting an algebraically closed proper subfield k and a multiplicative subgroup \(\Gamma \). 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 \((\Omega , k, \Gamma )\). This enables us to characterize the interpretable groups when \(\Gamma \) is divisible. Every interpretable group H in \((\Omega ,k, \Gamma )\) 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 \(\Gamma \) by an interpretable group N, which is the quotient of an algebraic group by a subgroup \(N_1\), 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 \(\Gamma \).