We show that the class of finitely generated virtually free groups is precisely the class of finitely generated demonstrable subgroups for Thompson’s group V. The class of demonstrable groups for V consists of all groups which can embed into V with a natural dynamical behavior 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 Thompson’s group V is a candidate as a universal co&#x1d49e;ℱ group by Lehnert’s conjecture, corresponding to the class of groups with context free co-word problem (as introduced by Holt, Rees, Rover, and Thomas). Our main results answers a question of Berns-Zieve, Fry, Gillings, Hoganson, and Matthews, and separately of Bleak and Salazar-Diaz, and it fits into the larger exploration of the class of co&#x1d49e;ℱ groups as it shows that all four of the known closure properties of the class of co&#x1d49e;ℱ groups hold for the set of finitely generated subgroups of V.
['Daniel Bennett', 'Collin Bleak']
A dynamical definition of f.g. virtually free groups