OBSTRUCTIONS FOR SUBGROUPS OF THOMPSON’S GROUP V

JOSÉ BURILLO, SEAN CLEARY, AND CLAAS E. RÖVER

Abstract. Thompson’s group V has a rich variety of subgroups, containing all finite groups, all finitely generated free groups and all finitely generated abelian groups, the finitary permutation group of a countable set, as well as many wreath products and other families of groups. Here, we describe some obstructions for a given group to be a subgroup of V.

Thompson constructed a finitely presented group now known as V as an early example of a finitely presented infinite simple group. The group V contains a remarkable variety of subgroups, such as the finitary infinite permutation group S_{∞}, and hence all (countable locally) finite groups, finitely generated free groups, finitely generated abelian groups, Houghton’s groups, copies of Thompson’s groups F, T and V, and many of their generalizations. Moreover, the class of subgroups of V is closed under direct products and restricted wreath products with finite or infinite cyclic top group.

In this short survey, we summarize the development of properties of V focusing on those which prohibit various groups from occurring as subgroups of V.

Thompson’s group V has many descriptions. Here, we simply recall that V is the group of right-continuous bijections from the unit interval [0, 1] to itself, which map dyadic rational numbers to dyadic rational numbers, which are differentiable except at finitely many dyadic rational numbers, and with slopes, when defined, integer powers of 2. The elements of this group can be described by reduced tree pair diagrams of the type (S, T, π) where π is a bijection between the leaves of the two finite rooted binary trees S and T.

Higman [7] gave a different description of V, which he denoted as G_{2,1} in a family of groups generalizing V.

1. Obstructions

Higman [7] described several important properties of V which can serve as obstructions to subgroups occurring in V.

The authors are grateful for the hospitality of Durham University during the Symposium on Cohomological and Geometric Group Theory. The first author acknowledges support from MEC grant MTM2011–25955. The second author acknowledges support from the National Science Foundation and that this work was partially supported by a grant from the Simons Foundation (#234548 to Sean Cleary).

Theorem 1.1 ([7]). An element of infinite order in V has only finitely many roots.

This prevents all Baumslag-Solitar groups B_{m,n} = ⟨a, b | a^{n}b = ba^{m}⟩ from occurring as subgroups of V, if m divides n; see [11].

Theorem 1.2 ([7]). Torsion free abelian subgroups of V are free abelian, and their centralizers have finite index in their normalizers in V.

This prevents GL_{n}(ℤ) from occurring as a subgroup of V for n ≥ 2.

A group is torsion locally finite if every torsion subgroup is locally finite. That is, if every finitely generated torsion subgroup is finite. Röver [10] showed

Theorem 1.3 ([10]). Thompson’s group V is torsion locally finite.

This rules out many branch groups from occurring as subgroups of V, including the Grigorchuk groups of intermediate growth [5], the Gupta-Sidki groups [6], and also rules out Burnside groups.

Holt and Röver [8] showed that V has indexed co-word problem.

Theorem 1.4 ([8]). The set of words (over an arbitrary but fixed finite generating set) which do not represent the identity in V is an indexed language, and hence can be recognized by a nested-stack automaton.

This property is not easy to verify, however. Groups which do not have an indexed co-word problem cannot thus occur as a subgroup of V.

Lehnert and Schweitzer [9] improved this result.

Theorem 1.5 ([9]). The set of words (over an arbitrary but fixed finite generating set) which do not represent the identity in V is a context-free language, and hence can be recognized by a pushdown automaton.

Again, this property is inherited by finitely generated subgroups, but the condition is still not easy to verify.

One theorem of Higman [7] together with a metric estimate of Birget [1] gives another obstruction.

Theorem 1.6 ([7]). For any element v of infinite order in V, there is a power v^{n} such that for the reduced tree pair diagram (S, T) for v^{n}, there is a leaf i in the source tree S which is paired with a leaf j in the target tree T where the location of j is a child of the location of i.

Theorem 1.7 ([1]). For any finite generating set of V, There are constants C and C′ such that word length |v| of an element of V with respect to that generating set satisfies Cn ≤ |v| ≤ C′n log n where n is the size of the reduced tree pair diagram representing v.

Since the powers of v^{n} will have length thus growing linearly, these two theorems give as a consequence the following.

Theorem 1.8. Cyclic subgroups of V are undistorted.

We note that this argument applies as well to generalizations of V where there is a linear lower bound on word length in terms of the number of carets, such as braided versions of V [4].

This last theorem has an obvious corollary.

Corollary 1.9. If a group embeds in V, its cyclic subgroups must be undistorted.

The reason for this is that in a chain of subgroups G ⊃ H ⊃ K the distortion of K in H cannot be larger than the distortion of K in G.

We note that Bleak, Bowman, Gordon, Graham, Hughes, Matucci and J. Sapir [3] used Brin’s methods of revealing pairs for elements of V to show that cyclic subgroups of V are undistorted.

This result excludes all Baumslag-Solitar groups with |n| ≠ |m|, as these have distorted cyclic subgroups. An alternative argument excluding these groups is due to Bleak, Matucci and Neunhöffer [2].

References

[1] Jean-Camille Birget. The groups of Richard Thompson and complexity. Internat. J. Algebra Comput., 14(5-6):569–626, 2004. International Conference on Semigroups and Groups in honor of the 65th birthday of Prof. John Rhodes.

[2] C. Bleak, F. Matucci, and M. Neunhöffer. Embeddings into Thompson’s group V and coCF groups. arXiv e-prints, December 2013.

[3] Collin Bleak, Hannah Bowman, Alison Gordon, Garrett Graham, Jacob Hughes, Francesco Matucci, and Jenya Sapir. Centralizers in R. Thompson’s group V_{n}, 2011.

[4] José Burillo and Sean Cleary. Metric properties of braided Thompson’s groups. Indiana Univ. Math. J., 58(2):605–615, 2009.

[5] R. I. Grigorčuk. On Burnside’s problem on periodic groups. Funktsional. Anal. i Prilozhen., 14(1):53–54, 1980.

[6] Narain Gupta and Saïd Sidki. On the Burnside problem for periodic groups. Math. Z., 182(3):385–388, 1983.

[7] Graham Higman. Finitely presented infinite simple groups. Department of Pure Mathematics, Department of Mathematics, I.A.S. Australian National University, Canberra, 1974. Notes on Pure Mathematics, No. 8 (1974).

[8] Derek F. Holt and Claas E. Röver. Groups with indexed co-word problem. Internat. J. Algebra Comput., 16(5):985–1014, 2006.

[9] J. Lehnert and P. Schweitzer. The co-word problem for the Higman-Thompson group is context-free. Bull. Lond. Math. Soc., 39(2):235–241, 2007.

[10] Claas E. Röver. Constructing finitely presented simple groups that contain Grigorchuk groups. J. Algebra, 220(1):284–313, 1999.

[11] Claas E. Röver. Subgroups of finitely presented simple groups. PhD thesis, University of Oxford, 1999.

Departament de Matemàtica Aplicada IV, Universitat Politècnica de Catalunya, Escola Politècnica Superior de Castelldefels, Esteve Terrades 5, 08860 Castelldefels (Barcelona), Spain

E-mail address: burillo@ma4.upc.edu

Department of Mathematics, The City College of New York, NY, USA

E-mail address: cleary@sci.ccny.cuny.edu

School of Mathematics, Statistics and Applied Mathematics, National University of Ireland Galway, University Road, Galway, Ireland

E-mail address: claas.roever@nuigalway.ie