101. A dynamical definition of f.g. virtually free groups
- Author
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Bennett, Daniel and Bleak, Collin
- Subjects
Mathematics - Group Theory ,Computer Science - Formal Languages and Automata Theory ,Mathematics - Dynamical Systems ,20F10, 20E06, 68Q45, 37B05 - 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 $V.$, Comment: 13 pp, 6 figures
- Published
- 2015
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