On Definable Galois Groups and the Strong Canonical Base Property

In [3], 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 [3], and prove its 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.