Connected components of definable groups and o-minimality I

TL;DR

This paper explores the structure of definable groups in o-minimal theories, providing examples of groups with distinct connected components and linking definable amenability to bounded orbits, thus advancing understanding of model-theoretic properties.

Contribution

It introduces new examples of groups with different connected components and establishes a link between definable amenability and bounded orbits in o-minimal structures.

Findings

G^00 differs from G^000 in certain groups

Definably amenable groups have bounded orbits in o-minimal structures

New non G-compact first order theories are constructed

Abstract

We give examples of groups G such that G^00 is different from G^000. We also prove that for groups G definable in an o-minimal structure, G has a "bounded orbit" iff G is definably amenable. These results answer questions of Gismatullin, Newelski, Petrykovski. The examples also give new non G-compact first order theories.