On non-abelian dp-minimal groups

TL;DR

This paper explores the structure of dp-minimal groups, showing torsion-free groups are abelian, those with a distal f-generic type are virtually nilpotent, and groups with the uniform chain condition are virtually solvable.

Contribution

It establishes new structural results for dp-minimal groups under various hypotheses, including conditions for abelianness, nilpotency, and solvability.

Findings

Torsion-free dp-minimal groups are abelian.

Groups with a distal f-generic type are virtually nilpotent.

Groups with the uniform chain condition are virtually solvable.

Abstract

Let $G$ be a dp-minimal group; we prove some consequences of several different hypotheses on $G$. First, if $G$ is torsion-free, then it is abelian. Second, if $G$ admits a distal f-generic type, then it is virtually nilpotent; we prove this by equipping the quotient of $G$ by its FC-center in this case with a valued group structure. Finally, if $G$ has the uniform chain condition, for example if $G$ is stable, then $G$ is virtually solvable.