A Galois correspondence for automorphism groups of structures with the Lascar Property

TL;DR

This paper introduces the Lascar Property to analyze automorphism groups of countable structures, establishing a Galois correspondence and characterizing automorphism groups in various theories, extending prior results.

Contribution

It generalizes the Galois correspondence for automorphism groups using the Lascar Property, unifying and extending previous analyses for different classes of structures.

Findings

Established a definable Galois correspondence under the Lascar Property.

Characterized automorphism groups for models of ACF₀, DCF₀, and infinite K-vector spaces.

Unified analysis for ω-categorical and other structures with weak elimination of imaginaries.

Abstract

Generalizing the $\omega$-categorical context, we introduce a notion, which we call the Lascar Property, that allows for a fine analysis of the topological isomorphisms between automorphism groups of countable structures satisfying this property. In particular, under the assumption of the Lascar Property, we exhibit a definable Galois correspondence between pointwise stabilizers of finitely generated Galois algebraically closed subsets of $M$ and finitely generated Galois algebraically closed subsets of $M$. We use this to characterize the group of automorphisms of $\mathrm{Aut}(M)$, for $M$ the countable saturated model of $\mathrm{ACF}_0$, $\mathrm{DCF}_0$, or the theory of infinite $\mathrm{K}$-vector spaces, generalizing results of Evans $\&$ Lascar, and Konnerth, while at the same time subsuming the analysis from [11] for $\omega$-categorical structures with weak elimination of imaginaries.