# Nilpotent groups, o-minimal Euler characteristic, and linear algebraic groups

## TL;DR

This paper reveals a deep connection between o-minimal definable groups and linear algebraic groups, showing that o-minimal nilpotent groups have algebraic decompositions and characterizing nilpotent Lie groups definable in o-minimal structures.

## Contribution

It establishes a correspondence between o-minimal definable nilpotent groups and linear algebraic groups, including algebraic decompositions and a characterization of nilpotent Lie groups.

## Key findings

- Nilpotency in o-minimal groups is equivalent to the normalizer property.
- O-minimal nilpotent groups admit algebraic decompositions.
- A nilpotent Lie group is o-minimally definable iff it is linear algebraic.

## Abstract

We establish a surprising correspondence between groups definable in o-minimal structures and linear algebraic groups, in the nilpotent case. It turns out that in the o-minimal context, like for finite groups, nilpotency is equivalent to the normalizer property or to uniqueness of Sylow subgroups. As a consequence, we show algebraic decompositions of o-minimal nilpotent groups, and we prove that a nilpotent Lie group is definable in an o-minimal expansion of the reals if and only if it is a linear algebraic group.