  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 $V.$
