Subdirect products of finitely presented metabelian groups

TL;DR

This paper investigates the finite presentability of subdirect products of finitely presented metabelian groups, showing that most such subdirect products are finitely presented and characterizing the exceptions via algebraic varieties.

Contribution

It demonstrates that, in a precise algebraic sense, most subdirect products of finitely presented metabelian groups are finitely presented, contrasting with the behavior in limit groups.

Findings

Most subdirect products are finitely presented.

Non-finitely presented cases form a lower-dimensional subvariety.

Provides an algebraic geometric framework for understanding these properties.

Abstract

There has been substantial investigation in recent years of subdirect products of limit groups and their finite presentability and homological finiteness properties. To contrast the results obtained for limit groups, Baumslag, Bridson, Holt and Miller investigated subdirect products (fibre products) of finitely presented metabelian groups. They showed that, in contrast to the case for limit groups, such subdirect products could have diverse behaviour with respect to finite presentability. We show that, in a sense that can be made precise, `most' subdirect products of a finite set of finitely presented metabelian groups are again finitely presented. To be a little more precise, we assign to each subdirect product a point of an algebraic variety and show that, in most cases, those points which correspond to non-finitely presented subdirect products form a subvariety of smaller dimension.