Interpretable groups in Mann pairs

In this paper, we study an algebraically closed field $$\Omega $$Ω expanded by two unary predicates denoting an algebraically closed proper subfield k and a multiplicative subgroup $$\Gamma $$Γ. 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 $$(\Omega , k, \Gamma )$$(Ω,k,Γ). This enables us to characterize the interpretable groups when $$\Gamma $$Γ is divisible. Every interpretable group H in $$(\Omega ,k, \Gamma )$$(Ω,k,Γ) 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 $$\Gamma $$Γ by an interpretable group N, which is the quotient of an algebraic group by a subgroup $$N_1$$N1, 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 $$\Gamma $$Γ.