Definable groups as homomorphic images of semilinear and field-definable groups

TL;DR

This paper studies definably compact groups in o-minimal structures, showing they can be decomposed into semi-linear and field-definable parts, and proves the Compact Domination Conjecture in this setting.

Contribution

It provides a structure theorem for definably compact groups in o-minimal expansions, linking them to semi-linear and real closed field definable groups, and proves the Compact Domination Conjecture.

Findings

Decomposition of definably compact groups into semi-linear and field-definable components

Structure theorems for locally definable covers of these groups

Proof of the Compact Domination Conjecture in o-minimal expansions

Abstract

We analyze definably compact groups in o-minimal expansions of ordered groups as a combination of semi-linear groups and groups definable in o-minimal expansions of real closed fields. The analysis involves structure theorems about their locally definable covers. As a corollary, we prove the Compact Domination Conjecture in o-minimal expansions of ordered groups.