Smoothness of bounded invariant equivalence relations

TL;DR

This paper extends key theorems on the smoothness of bounded invariant equivalence relations, explores their properties, and proves a conjecture, thereby broadening understanding in model theory.

Contribution

It generalizes main theorems to a wider class of relations, clarifies the relationship between properties like smoothness and type-definability, and proves a conjecture regarding connected components.

Findings

Generalized theorems to broader classes of relations

Established equivalences between properties of invariant relations

Proved the necessity of a key technical assumption in a conjecture

Abstract

We generalise the main theorems from the paper "The Borel cardinality of Lascar strong types" by I. Kaplan, B. Miller and P. Simon to a wider class of bounded invariant equivalence relations. We apply them to describe relationships between fundamental properties of bounded invariant equivalence relations (such as smoothness or type-definability) which also requires finding a series of counterexamples. Finally, we apply the generalisation mentioned above to prove a conjecture from a paper by the first author and J. Gismatullin, showing that the key technical assumption of the main theorem (concerning connected components in definable group extensions) from that paper is not only sufficient but also necessary to get the conclusion.