Groups with poly-context-free word problem

TL;DR

This paper investigates groups with word problems that are intersections of finitely many context-free languages, establishing their relation to virtually finitely generated subgroups of direct products of free groups and exploring soluble groups.

Contribution

It proves that certain groups have poly-context-free word problems and conjectures a characterization involving virtually finitely generated subgroups of direct products of free groups.

Findings

Virtually finitely generated subgroups of direct products of free groups have poly-context-free word problems.

The conjecture holds for several classes of soluble groups, including metabelian and torsion-free soluble groups.

Introduces techniques for proving languages are not poly-context-free, which may be of independent interest.

Abstract

We consider the class of groups whose word problem is poly-context-free; that is, an intersection of finitely many context-free languages. We show that any group which is virtually a finitely generated subgroup of a direct product of free groups has poly-context-free word problem, and conjecture that the converse also holds. We prove our conjecture for several classes of soluble groups, including metabelian groups and torsion-free soluble groups, and present progress towards resolving the conjecture for soluble groups in general. Some of the techniques introduced for proving languages not to be poly-context-free may be of independent interest.