On definable Galois groups and the strong canonical base property

TL;DR

This paper establishes an equivalence between the strong canonical base property and the rigidity of definable Galois groups within a broader finite rank setting, extending previous results on 1-based groups.

Contribution

It provides a more general definition of the strong canonical base property and proves its equivalence to Galois group rigidity, broadening the scope of earlier work.

Findings

Strong canonical base property is equivalent to Galois group rigidity.

The paper generalizes the context from previous finite rank environments.

It elaborates on the relationship between 1-based groups and rigidity.

Abstract

In \cite{HPP}, Hrushovski and the authors proved, in a certain finite rank environment, that rigidity of definable Galois groups implies that $T$ has the canonical base property in a strong form, " internality to" being replaced by "algebraicity in". In the current paper we give a reasonably robust definition of the "strong canonical base property" in a rather more general finite rank context than \cite{HPP}, and prove its {\em equivalence} with rigidity of the relevant definable Galois groups. The new direction is an elaboration on the old result that $1$-based groups are rigid.