EDT0L solutions to equations in group extensions

TL;DR

This paper explores the class of groups where solutions to systems of equations can be described by EDT0L languages, demonstrating closure properties and extending the class to include various complex groups, with solutions expressed via quasigeodesic normal forms.

Contribution

It establishes that the class of groups with EDT0L solution sets is closed under several group operations and extends this class to include groups with hyperbolic components and virtually abelian groups.

Findings

Closure under direct products and wreath products with finite groups

EDT0L solutions in groups containing hyperbolic group products as finite index subgroups

Solutions in virtually abelian groups with rational constraints

Abstract

We show that the class of groups where EDT0L languages can be used to describe solution sets to systems of equations is closed under direct products, wreath products with finite groups, and passing to finite index subgroups. We also add the class of groups that contain a direct product of hyperbolic groups as a finite index subgroup to the list of groups where solutions to systems of equations can be expressed as an EDT0L language. This includes dihedral Artin groups. We also show that the systems of equations with rational constraints in virtually abelian groups have EDT0L solutions, and the addition of recognisable contraints to any system preserves the property of having EDT0L solutions. These EDT0L solutions are expressed with respect to quasigeodesic normal forms.