Galois actions of finitely generated groups rarely have model companions

2022/10/03 by Özlem Beyarslan, Beyarslan, Özlem, Piotr A. Kowalski +1
Mathematics · #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Logic (math.LO)

paper · pdf · doi:10.48550/arxiv.2210.00800

openalex publication_date 2022/10/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that if G is a finitely generated group such that its profinite completion \widehatG is ``far from being projective'' (that is the kernel of the universal Frattini cover of \widehatG is not a small profinite group), then the class of existentially closed G-actions on fields is not elementary. Since any infinite, finitely generated, virtually free, and not free group is ``far from being projective'', the main result of this paper corrects an error in our paper ``Model theory of fields with virtually free group actions'', Proc. London Math. Soc., (2) 118 (2019), 221--256 by showing the negation of Theorem 3.26 in that paper.

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