On the Cantor-Bendixson rank of metabelian groups

TL;DR

This paper investigates the Cantor-Bendixson rank of metabelian groups within the space of marked groups, revealing a sequence of finitely presented, virtually metabelian groups with ranks growing as omega^n.

Contribution

It introduces a sequence of finitely presented, virtually metabelian groups with Cantor-Bendixson ranks of omega^n, advancing understanding of the rank's behavior in this class.

Findings

Constructed a sequence of 2-generated, finitely presented, virtually metabelian groups

Demonstrated that their Cantor-Bendixson ranks are omega^n

Provides insights into the structure of the space of marked groups

Abstract

We study the Cantor-Bendixson rank of metabelian and virtually metabelian groups in the space of marked groups, and in particular, we exhibit a sequence (G_n) of 2-generated, finitely presented, virtually metabelian groups of Cantor-Bendixson rank omega^n.