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.

Local stability has been used in the recent years to treat problems in additive combinatorics. Whilst many of the techniques of geometric stability theory have been generalised to simple theories, there is no local treatment of simplicity. Kaplan and Shelah showed that the theory of the additive group of the integers together with a predicate for the prime integers is supersimple of rank 1, assuming Dickson’s conjecture. We will see how to use their result to deduce that all but finitely many integers belongs to infinitely many arithmetic progressions in the primes, which resonates with previous unconditional work (without assuming Dickson’s conjecture) of van der Corput and of Green. If times permits, we will discuss analogous results asymptotically for finite fields.