Lie-like decompositions of groups definable in o-minimal structures

TL;DR

This paper explores the structure of groups definable in o-minimal structures, revealing a Lie-like decomposition involving maximal 0-subgroups, and extends understanding of their analogies with real Lie groups.

Contribution

It introduces a novel decomposition framework for non-definably compact groups in o-minimal structures, highlighting the role of maximal 0-subgroups.

Findings

Existence of a unique maximal normal definable torsion-free subgroup

Decomposition of G/N into maximal definably compact and torsion-free parts

Definable homotopy equivalence of G to its maximal definably compact subgroup

Abstract

There are strong analogies between groups definable in o-minimal structures and real Lie groups. Nevertheless, unlike the real case, not every definable group has maximal definably compact subgroups. We study definable groups G which are not definably compact showing that they have a unique maximal normal definable torsion-free subgroup N; the quotient G/N always has maximal definably compact subgroups, and for every such a K there is a maximal definable torsion-free subgroup H such that G/N can be decomposed as G/N = KH, and the intersection between K and H is trivial. Thus G is definably homotopy equivalent to K. When G is solvable then G/N is already definably compact. In any case (even when G has no maximal definably compact subgroup) we find a definable Lie-like decomposition of G where the role of maximal tori is played by maximal 0-subgroups.