Visibly Pushdown Languages in Groups

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