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

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, provided the maximal normal torsion-free definable subgroup is nilpotent. As a consequence, we show definable 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 Lie isomorphic to a linear algebraic group.