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Preprint Number 1417

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1417. Krzysztof Krupiński, Tomasz Rzepecki
Galois groups as quotients of Polish groups
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Submission date: 24 April 2018

Abstract:

We present the (Lascar) Galois group of any countable theory as a quotient of a compact Polish group by an F_σ 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 ∅. As an easy conclusion of our main theorem, we get the main result from our recent paper joint with Anand Pillay, which says that for any strong type defined on a single complete type over ∅, 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.

Mathematics Subject Classification: 03C45, 54H20, 22C05, 03E15, 54H11

Keywords and phrases: topological dynamics, Galois groups, Polish groups, strong types,Borel cardinality, Rosenthal compacta.

Full text arXiv 1804.09247: pdf, ps.


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