This is joint work with Piotr Kowalski.

A G-field is a field, together with an action of a group G by field automorphisms. If an axiomatization for the class of existentially closed G-fields exists, we call the resulting theory G-TCF. If G is the group of integers, then G-TCF exists and it coincides with ACFA. More generally, when G is a finitely generated free group on n-generators, then the theory of existentially closed models is ACFA_n. It is also known that G-TCF exists if G is finite. We studied G-fields when G is a virtually free group, and showed that G-TCF exists.

I will summarize our previous work and present our current work in progress, concerning the existence of G-TCF for certain divisible abelian groups.