Visibly Pushdown Languages in Groups

2026-04-24

AI Summary

This study investigates when the word problem of a finitely generated group belongs to the class of visibly pushdown languages (VPLs). By integrating techniques from formal language theory, group theory, and automata theory, the authors establish that a group’s word problem is a VPL if and only if the group is finite. They further demonstrate that free reduction does not preserve the VPL property and prove the undecidability of solving equations in free groups under VPL constraints. Additionally, the paper formulates and provides evidence for the conjecture that the preimage of any VPL under the canonical map from the free monoid to the group is a recognizable subset. These results uncover a profound connection between VPLs and finite groups, thereby extending the applicability of formal language methods in group theory.

Abstract

In this paper we explore the connections between the class of Visibly Pushdown Languages ($\mathbf{VPL}$) and the natural sets of words one can associate to a finitely generated group. We show that the word problem of a finitely generated group is $\mathbf{VPL}$ exactly when the group is finite. We also show that free reduction does not preserve $\mathbf{VPL}$, and that finding solutions to equations in a free group with $\mathbf{VPL}$ constraints (as reduced words) is undecidable. We explore the structure of sets whose full preimage is $\mathbf{VPL}$, showing these are often recognisable sets. We conjecture that, in any group, this class is precisely the recognisable sets.

Problem

• Visibly Pushdown Languages
• word problem
• finitely generated groups
• free reduction
• recognisable sets

Innovation

• Visibly Pushdown Languages
• word problem
• finitely generated groups
• recognisable sets
• undecidability