A dynamical definition of f.g. virtually free groups

TL;DR

This paper characterizes finitely generated virtually free groups as exactly the demonstrable subgroups of Thompson's group V, linking group theory, dynamical systems, and formal language theory.

Contribution

It provides a dynamical characterization of finitely generated virtually free groups within Thompson's group V, connecting them to formal language classes and answering key open questions.

Findings

Finitely generated virtually free groups are exactly the demonstrable subgroups of V.

The class of groups with context-free word problem coincides with finitely generated virtually free groups.

Closure properties of the class of coCF groups hold for finitely generated subgroups of V.

Abstract

We show that the class of finitely generated virtually free groups is precisely the class of demonstrable subgroups for R. Thompson's group $V$. The class of demonstrable groups for $V$ consists of all groups which can embed into $V$ with a natural dynamical behaviour in their induced actions on the Cantor space $\mathfrak{C}_2 := \left\{0,1\right\}^\omega$. There are also connections with formal language theory, as the class of groups with context-free word problem is also the class of finitely generated virtually free groups, while R. Thompson's group $V$ is a candidate as a universal $co\mathcal{CF}$ group by Lehnert's conjecture, corresponding to the class of groups with context free co-word problem (as introduced by Holt, Rees, R\"over, and Thomas). Our main reults answers a question of Berns-Zieze, Fry, Gillings, Hoganson, and Matthews, and separately of Bleak and Salazar-D\'iaz, and it fits into the larger exploration of the class of $co\mathcal{CF}$ groups as it shows that all four of the known closure properties of the class of $co\mathcal{CF}$ groups hold for the set of finitely generated subgroups of $V.$