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

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 saturated structures satisfying this property. In particular, under these assumptions, we exhibit a Galois correspondence between pointwise stabilizers of finitely generated algebraically closed subsets of $M$ and finitely generated algebraically closed subsets of $M$. We use this to characterize the group of automorphisms of $\mathrm{Aut}(M)$, for $M$ a countable saturated model of ${\mathrm{ACF}}_0$ or an infinite-dimensional $\mathbb{K}$-vector space with $\mathbb{K}$ countable, generalizing a classical result of Evans $\&$ Lascar (1997), while at the same time subsuming the analysis of Paolini (2024) for $\omega$-categorical structures with weak elimination of imaginaries.