An o-minimalist view of the group configuration

TL;DR

This paper explores the group configuration in o-minimal structures, simplifying proofs and establishing new equivalences, while also addressing embedding questions for definably connected groups.

Contribution

It provides a simplified o-minimal proof of the group configuration theorem and introduces new equivalent formulations suitable for o-minimal settings.

Findings

Simplified proof of the group configuration in o-minimal structures

Formulation of equivalent versions in functional language and relations

Positive results on embedding definably connected groups into definable groups

Abstract

The group configuration in o-minimal structures gives rise, just like in the stable case, to a transitive action of a type-definable group on a partial type. Because $acl=dcl$ the o-minimal proof is significantly simpler than Hrushovski's original argument. Several equivalent versions, which are more suitable to the o-minimal setting, are formulated, in functional language and also in terms of a certain $4$-ary relation. In addition, the following question is considered: Can every definably connected type-definable group be definably embedded into a definable group of the same dimension? Two simple cases with a positive answer are given.