Computing equations for residually free groups

TL;DR

This paper proves the non-existence of an algorithm to determine if the maximal residually free quotient of a finitely presented group is finitely presentable, and discusses explicit equations for subgroups of product of limit groups.

Contribution

It establishes the undecidability of certain properties of residually free groups and explores explicit descriptions of subgroups within products of limit groups.

Findings

No algorithm can decide if the maximal residually free quotient is finitely presentable.

Discusses the possibility of explicitly defining subgroups via equations.

Provides insights into the structure of residually free groups and their quotients.

Abstract

We show that there is no algorithm deciding whether the maximal residually free quotient of a given finitely presented group is finitely presentable or not. Given a finitely generated subgroup G of a finite product of limit groups, we discuss the possibility of finding an explicit set of defining equations (i.e. of expressing G as the maximal residually free quotient of an explicit finitely presented group).