Connected components of definable groups, and o-minimality II

TL;DR

This paper investigates the structure of connected components in definable groups within o-minimal structures, revealing their relation to Lie groups and universal covers, thus advancing understanding of definable group topology.

Contribution

It characterizes the quotient G^00/G^000 as a quotient of a connected compact Lie group by a dense finitely generated subgroup, linking model theory and Lie group theory.

Findings

G^00/G^000 is a quotient of a compact Lie group by a dense subgroup

Universal covers of semisimple Lie groups play a key role

Provides a structural description of connected components in definable groups

Abstract

We study the connected components G^00, G^000 and their quotients for a group G definable in a saturated o-minimal expansion of a real closed field. We show that G^00/G^000 is naturally the quotient of a connected compact commutative Lie group by a dense finitely generated subgroup. We also highlight the role of universal covers of semisimple Lie groups.