Localization, metabelian groups, and the isomorphism problem

TL;DR

This paper explores the isomorphism problem for finitely generated residually nilpotent metabelian groups, introducing the para-G relation, and demonstrates methods to distinguish groups within this class using localization and algebraic techniques.

Contribution

It introduces the para-G relation as a coarser equivalence for metabelian groups and provides methods to solve the isomorphism problem for certain classes of these groups.

Findings

The para-G relation is an equivalence relation.

Some equivalence classes contain only one isomorphism class.

The sequence of torsion-free ranks of lower central quotients is computable.

Abstract

If G and H are finitely generated, residually nilpotent metabelian groups, H is termed para-G if there is a homomorphism of G into H which induces an isomorphism between the corresponding terms of their lower central quotient groups. We prove that this is an equivalence relation. It is a much coarser relation than isomorphism, our ultimate concern. It turns out that many of the groups in a given equivalence class share various properties including finite presentability. There are examples, such as the lamplighter group, where an equivalence class consists of a single isomorphism class and others where this is not the case. We give several examples where we solve the Isomorphism Problem. We prove also that the sequence of torsion-free ranks of the lower central quotients of a finitely generated metabelian group is computable. In a future paper we plan on proving that there is an algorithm to compute the numerator and denominator of the rational Poincar\'e series of a finitely generated metabelian group and will carry out this computation in a number of examples, which may shed a tiny bit of light on the Isomorphism Problem. Our proofs use localization, class field theory and some constructive commutative algebra.