Bounded Invariant Equivalence Relations

TL;DR

This paper investigates strong types and Galois groups in model theory using topological and descriptive-set-theoretical methods, providing a general framework that extends and improves previous results on their cardinalities and structures.

Contribution

It offers an abstract, non-model-theoretic approach to analyze strong types and Galois groups, revealing new insights into their topological and set-theoretic properties.

Findings

Strong type spaces are locally quotients of compact Polish groups.

Strong types are smooth iff they are type-definable.

Quotients of type-definable groups by analytic subgroups are either finite or have continuum cardinality.

Abstract

We study strong types and Galois groups in model theory from a topological and descriptive-set-theoretical point of view, leaning heavily on topological dynamical tools. More precisely, we give an abstract (not model theoretic) treatment of problems related to cardinality and Borel cardinality of strong types, quotients of definable groups and related objecets, generalising (and often improving) essentially all hitherto known results in this area. In particular, we show that under reasonable assumptions, strong type spaces are "locally" quotients of compact Polish groups. It follows that they are smooth if and only if they are type-definable, and that a quotient of a type-definable group by an analytic subgroup is either finite or of cardinality at least continuum.