Residually rationally solvable one-relator groups

TL;DR

This paper investigates the structure of one-relator groups, showing their rational derived series has specific properties and providing a precise characterization of when such groups are residually rationally solvable.

Contribution

It introduces a detailed analysis of the rational derived series in one-relator groups and characterizes residual rational solvability.

Findings

The intersection of the rational derived series is rationally perfect.

This intersection is normally generated by a single element.

Provides a precise criterion for residual rational solvability.

Abstract

We show that the intersection of the rational derived series of a one-relator group is rationally perfect and is normally generated by a single element. As a corollary, we characterise precisely when a one-relator group is residually rationally solvable.