A dynamical definition of f.g. virtually free groups

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 C₂ := {0,1}^ω. 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 coCF group by Lehnert's conjecture, corresponding to the class of
groups with context free co-word problem (as introduced by Holt, Rees, Röver, and Thomas). Our main results answers a question of Berns-Zieze, Fry, Gillings, Hoganson, and Matthews, and separately of Bleak and Salazar-Días, and fits into the larger exploration of the class of coCF groups as it shows that all four of the known properties of the class of coCF groups hold for the set of finitely generation subgroups of V.