On non-abelian dp-minimal groups I: the torsion-free and distal cases

Abstract

We give some results on dp-minimal groups. First we show that any torsion-free dp-minimal group is abelian; along the way show that any dp-minimal group admitting a principal f-generic type whose realizations are non-torsion is nilpotent-by-finite of class at most $2$. We then investigate the question of whether *every* dp-minimal group $G$ is nilpotent-by-finite. There are naturally two cases: either (1) $G$ admits a distal f-generic type or (2) $G$ admits a generically stable f-generic type. In this paper we resolve case (1). This follows from a more general structural analysis, in which, assuming that $G$ admits a distal f-generic type, we show that the quotient of $G$ by its FC-center can be naturally equipped with the structure of a valued group; we then use this valuation structure to show that indeed $G$ is nilpotent-by-finite. Case (2) of the question will be resolved in an upcoming joint paper with Eran Alouf and Frank Wagner.

arXiv commentremoved the section on ucc groups, some material of which will be needed in an upcoming joint paper with Eran Alouf and Frank Wagner where we will resolve case (2). changed copyright license to prohibit commercial use