Interpretable groups in Mann pairs.

In this paper, we study an algebraically closed field Ω<inline-graphic></inline-graphic> expanded by two unary predicates denoting an algebraically closed proper subfield <italic>k</italic> and a multiplicative subgroup Γ<inline-graphic></inline-graphic>. 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 (Ω,k,Γ)<inline-graphic></inline-graphic>. This enables us to characterize the interpretable groups when Γ<inline-graphic></inline-graphic> is divisible. Every interpretable group <italic>H</italic> in (Ω,k,Γ)<inline-graphic></inline-graphic> is, up to isogeny, an extension of a direct sum of <italic>k</italic>-rational points of an algebraic group defined over <italic>k</italic> and an interpretable abelian group in Γ<inline-graphic></inline-graphic> by an interpretable group <italic>N</italic>, which is the quotient of an algebraic group by a subgroup N1<inline-graphic></inline-graphic>, which in turn is isogenous to a cartesian product of <italic>k</italic>-rational points of an algebraic group defined over <italic>k</italic> and an interpretable abelian group in Γ<inline-graphic></inline-graphic>. [ABSTRACT FROM AUTHOR]