Groups definable in o-minimal structures: structure theorem, G^000, definable amenability and bounded orbits

TL;DR

This paper advances the understanding of definable groups in o-minimal structures by resolving key open problems, including the equality of G^00 and G^000, and linking definable amenability to bounded orbits.

Contribution

It proves the equality of G^00 and G^000 in o-minimal groups and establishes the equivalence between definable amenability and the existence of a type with bounded orbit.

Findings

G^00 equals G^000 in o-minimal groups

Definable amenability is equivalent to having a type with bounded orbit

Almost exactness of the G to G^00 functor in this context

Abstract

We settle some open problems in the special case of groups in o-minimal structures, such as the equality of G^00 and G^000 and the equivalence of definable amenability and existence of a type with bounded orbit. We prove almost exactness of the G to G^00 functor. We ask further questions about types with bounded orbits in NIP theories.