A dynamical definition of f.g. virtually free groups

Abstract

We show that the class of finitely generated virtually free groups is precisely the class of finitely generated demonstrable subgroups for Thompson’s group [Formula: see text]. The class of demonstrable groups for [Formula: see text] consists of all groups which can embed into [Formula: see text] with a natural dynamical behavior in their induced actions on the Cantor space [Formula: see text]. 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 [Formula: see text] is a candidate as a universal [Formula: see text] 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-Zieve, Fry, Gillings, Hoganson, and Matthews, and separately of Bleak and Salazar-Díaz, and it fits into the larger exploration of the class of [Formula: see text] groups as it shows that all four of the known closure properties of the class of [Formula: see text] groups hold for the set of finitely generated subgroups of [Formula: see text].