Model theory of fields with virtually free group actions

Abstract

For a group G, we define the notion of a G‐kernel and show that the properties of G‐kernels are closely related with the existence of a model companion of the theory of Galois actions of G. Using Bass–Serre theory, we show that this model companion exists for virtually free groups generalizing the existing results about free groups and finite groups. We show that the new theories we obtain are not simple and not even NTP2.

Model theory of fields with free operators in positive characteristic

“…We can prove now the main criterion about prolonging B-kernels to B-regular realizations. It is analogous to and plays the same role as Kernel-Prolongation Lemmas from [2] (e.g. Lemma 2.13, Proposition 3.7 or Proposition 3.17 in [2]).…”

“…It is analogous to and plays the same role as Kernel-Prolongation Lemmas from [2] (e.g. Lemma 2.13, Proposition 3.7 or Proposition 3.17 in [2]). We advice the reader to recall the definition of the K-subvariety E ⊆ τ ∂ (W ) and the morphism π E , which appear before Corollary 2.19.…”

“…The fact that C is separably closed follows from Lemma 2.7(2) (separable algebraic field extensions areétale). To show that C has an infinite imperfection degree, it is enough to show (using Proposition 2.20, Lemma 2.7(1), Lemma 3.1 (2) and Corollary 4.5) that for each n > 0, there is a B-field structure ∂ on the field of rational functions k(X 1 , . .…”

Model theory of Galois actions of torsion Abelian groups

“…More recently, Daniel Hoffmann and the second author considered in [8] the case of finite groups (being unaware then of Sjögren's work from [20]). In [1], the authors of this paper extended some of the results from [3] and [8] into a very natural common context of virtually free groups. This work is a continuation of the general line of research from [1], however, it goes in a different direction, that is we consider infinite torsion Abelian groups.…”

“…In [1], the authors of this paper extended some of the results from [3] and [8] into a very natural common context of virtually free groups. This work is a continuation of the general line of research from [1], however, it goes in a different direction, that is we consider infinite torsion Abelian groups. Let A be a torsion Abelian group.…”