Some torsion-free solvable groups with few subquotients

TL;DR

This paper constructs specific torsion-free solvable groups with infinite rank that have very limited metabelian subquotients, challenging existing theorems about solvable groups.

Contribution

It introduces finitely generated torsion-free solvable groups with infinite rank but only virtually abelian metabelian subquotients, showing a limitation of Kropholler's theorem.

Findings

Existence of torsion-free solvable groups with infinite rank and limited subquotients

All finitely generated metabelian subgroups are virtually abelian

Counterexample to a torsion-free analogue of Kropholler's theorem

Abstract

We construct finitely generated torsion-free solvable groups $G$ that have infinite rank, but such that all finitely generated torsion-free metabelian subquotients of $G$ are virtually abelian. In particular all finitely generated metabelian subgroups of $G$ are virtually abelian. The existence of such groups shows that there is no "torsion-free version" of P. Kropholler's theorem, which characterises solvable groups of infinite rank via their metabelian subquotients.