We revisit the model theory of fields with a group action by automorphisms, focusing on the existence of the model companion G-TCF. We explain a flaw in earlier work and present the corrected result: for finitely generated virtually-free groups G, G-TCF exists if and only if G is finite or free. This is joint work with Piotr Kowalski.

The assertion belongs to the representation theory of partially ordered sets, to Non-Hausdorff topology and to domain theory, but is (co-)motivated by model theoretic questions about the analysis of structures that can be seen as global sections of a sheaf (like a ring or like a generalized product in the Feferman-Vaught theorem). I will first explain my interest in the statement of the title and then construct the asserted space in a functorial way.