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

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 $\mathrm{A}\mathrm{u}\mathrm{t}(M)$, for $M$ the countable saturated model of ${\mathrm{A}\mathrm{C}\mathrm{F}}_0$, ${\mathrm{D}\mathrm{C}\mathrm{F}}_0$, or the theory of infinite $\mathrm{K}$-vector spaces, generalizing results of Evans $\mathrm{\&}$ Lascar, and Konnerth, while at the same time subsuming the analysis from [11] for $\omega$-categorical structures with weak elimination of imaginaries.