turkmath.org

This is joint work with Özlem Beyarslan. For a fixed group G, we consider the actions of G on fields by field automorphisms. There is a very general notion (which I will explain) of an '' existentially closed '' action, which for the trivial group corresponds just to an algebraically closed field. It is an open and interesting problem, for which groups G, the class of such existentially closed actions is axiomatizable, that is a " model companion exists ". Generalizing the known results about finite and free groups, we showed (using Bass–Serre theory) that a model companion exists for finitely generated virtually free groups (Proc. London Math. Soc. (3) 118 (2019) 221–256). Time permitting, I will also discuss our work in progress about actions of divisible Abelian torsion groups.