Galois groups as quotients of Polish groups

TL;DR

This paper characterizes the Lascar Galois group of countable theories as a quotient of a compact Polish group, linking topological group structures with model-theoretic properties like smoothness and type-definability.

Contribution

It provides a new topological description of Galois groups as quotients of Polish groups, extending to strong types and bounded quotients of type-definable groups.

Findings

Galois groups as quotients of Polish groups by $F_\sigma$ subgroups

Smoothness of strong types is equivalent to type-definability

Generalization to bounded quotients of type-definable groups

Abstract

We present the (Lascar) Galois group of any countable theory as a quotient of a compact Polish group by an $F_\sigma$ normal subgroup: in general, as a topological group, and under NIP, also in terms of Borel cardinality. This allows us to obtain similar results for arbitrary strong types defined on a single complete type over $\emptyset$. As an easy conclusion of our main theorem, we get the main result from our recent paper joint with Andand Pillay, which says that for any strong type defined on a single complete type over $\emptyset$, smoothness is equivalent to type-definability. We also explain how similar results are obtained in the case of bounded quotients of type-definable groups. This gives us a generalization of a former result from the aforementioned paper about bounded quotients of type-definable subgroups of definable groups.