Packing subgroups in solvable groups

TL;DR

This paper investigates the bounded packing property in various classes of solvable groups, establishing positive results for certain subgroups and providing counterexamples for others, thereby clarifying the scope of the property.

Contribution

It proves that many subgroups of solvable groups satisfy bounded packing, but also constructs a counterexample showing the property does not hold universally.

Findings

Virtually nilpotent-by-polycyclic groups satisfy bounded packing.

Metabelian and linear solvable groups also satisfy bounded packing.

Counterexample of a solvable group of derived length 3 lacking bounded packing.

Abstract

We show that any subgroup of a (virtually) nilpotent-by-polycyclic group satisfies the bounded packing property of Hruska-Wise. In particular, the same is true about metabelian groups and linear solvable groups. However, we find an example of a finitely generated solvable group of derived length 3 which admits a finitely generated subgroup without the bounded packing property. In this example the subgroup is a metabelian retract also. Thus we obtain a negative answer to Problem 2.27 of Hruska-Wise. On the other hand, we show that polycyclic subgroups of solvable groups satisfy the bounded packing property.