# Thompson groups

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Three groups

This article is about the three unusual infinite groups  F, T and V found by Richard Thompson. For the sporadic simple group found by John G. Thompson, see [Thompson sporadic group](/source/Thompson_sporadic_group).

In [mathematics](/source/Mathematics), the **Thompson groups** (also called **Thompson's groups**, **vagabond groups** or **chameleon groups**) are three [groups](/source/Group_(mathematics)), commonly denoted F ⊆ T ⊆ V {\displaystyle F\subseteq T\subseteq V} , that were introduced by Richard Thompson in some unpublished handwritten notes in 1965 as a possible counterexample to the [von Neumann conjecture](/source/Von_Neumann_conjecture). Of the three, *F* is the most widely studied, and is sometimes referred to as **the Thompson group** or **Thompson's group**.

The Thompson groups, and *F* in particular, have a collection of unusual properties that have made them counterexamples to many general conjectures in group theory. All three Thompson groups are infinite but [finitely presented](/source/Finitely_presented_group). The groups *T* and *V* are (rare) examples of infinite but finitely-presented [simple groups](/source/Simple_group). The group *F* is not simple but its [derived subgroup](/source/Derived_subgroup) [*F*,*F*] is and the quotient of *F* by its derived subgroup is the [free abelian group](/source/Free_abelian_group) of rank 2. *F* is [totally ordered](/source/Totally_ordered_group), has [exponential growth](/source/Growth_rate_(group_theory)), and does not contain a [subgroup](/source/Subgroup) isomorphic to the [free group](/source/Free_group) of rank 2.

It is conjectured that *F* is not [amenable](/source/Amenable_group) and hence a further counterexample to the long-standing but recently disproved [von Neumann conjecture](/source/Von_Neumann_conjecture) for finitely-presented groups: it is known that *F* is not [elementary amenable](/source/Elementary_amenable).

[Higman (1974)](#CITEREFHigman1974) introduced an infinite family of finitely presented simple groups, including Thompson's group *V* as a special case.

## Presentations

A finite [presentation](/source/Presentation_of_a_group) of *F* is given by the following expression:

- ⟨ A , B ∣ [ A B − 1 , A − 1 B A ] = [ A B − 1 , A − 2 B A 2 ] = i d ⟩ {\displaystyle \langle A,B\mid \ [AB^{-1},A^{-1}BA]=[AB^{-1},A^{-2}BA^{2}]=\mathrm {id} \rangle }

where [*x*,*y*] is the usual group theory [commutator](/source/Commutator#Group_theory), *xyx*−1*y*−1.

Although *F* has a finite presentation with 2 generators and 2 relations, it is most easily and intuitively described by the infinite presentation:

- ⟨ x 0 , x 1 , x 2 , … ∣ x k − 1 x n x k = x n + 1 f o r k < n ⟩ . {\displaystyle \langle x_{0},x_{1},x_{2},\dots \ \mid \ x_{k}^{-1}x_{n}x_{k}=x_{n+1}\ \mathrm {for} \ k<n\rangle .}

The two presentations are related by *x*0=*A*, *x**n* = *A*1−*n**BA**n*−1 for *n*>0.

## Other representations

The Thompson group *F* is generated by operations like this on  binary trees. Here *L* and *T* are nodes, but *A* *B* and *R* can be replaced by more general trees.

The group *F* also has realizations in terms of operations on ordered rooted [binary trees](/source/Binary_tree), and as a subgroup of the piecewise linear [homeomorphisms](/source/Homeomorphism) of the [unit interval](/source/Unit_interval) that preserve orientation and whose non-differentiable points are [dyadic rationals](/source/Dyadic_rational) and whose slopes are all powers of 2.

The group *F* can also be considered as acting on the [unit circle](/source/Unit_circle) by identifying the two endpoints of the unit interval, and the group *T* is then the group of automorphisms of the unit circle obtained by adding the homeomorphism *x*→*x*+1/2 mod 1 to *F*. On binary trees this corresponds to exchanging the two trees below the root. The group *V* is obtained from *T* by adding the discontinuous map that fixes the points of the half-open interval [0,1/2) and exchanges [1/2,3/4) and [3/4,1) in the obvious way. On binary trees this corresponds to exchanging the two trees below the right-hand descendant of the root (if it exists).

The Thompson group *F* is the group of order-preserving automorphisms of the free [Jónsson–Tarski algebra](/source/J%C3%B3nsson%E2%80%93Tarski_algebra) on one generator.

## Amenability

The conjecture of Thompson that *F* is not [amenable](/source/Amenable_group) was further popularized by R. Geoghegan—see also the Cannon–Floyd–Parry article cited in the references below. Its current status is open: E. Shavgulidze[1] published a paper in 2009 in which he claimed to prove that *F* is amenable, but an error was found, as is explained in the [MR](/source/Mathematical_Reviews) review.

It is known that *F* is not [elementary amenable](/source/Elementary_amenable), see Theorem 4.10 in Cannon–Floyd–Parry.

If *F* is **not** amenable, then it would be another counterexample to the now disproved [von Neumann conjecture](/source/Von_Neumann_conjecture) for finitely-presented groups, which states that a finitely-presented group is amenable [if and only if](/source/If_and_only_if) it does not contain a copy of the free group of rank 2.

## Connections with topology

The group *F* was rediscovered at least twice by topologists during the 1970s. In a paper that was only published much later but was in circulation as a preprint at that time, [P. Freyd](/source/Peter_Freyd) and A. Heller [2] showed that the *shift map* on *F* induces an unsplittable [homotopy](/source/Homotopy) idempotent on the [Eilenberg–MacLane space](/source/Eilenberg%E2%80%93MacLane_space) *K(F,1)* and that this is universal in an interesting sense. This is explained in detail in Geoghegan's book (see references below). Independently, J. Dydak and P. Minc [3] created a less well-known model of *F* in connection with a problem in shape theory.

In 1979, R. Geoghegan made four conjectures about *F*: (1) *F* has [type **FP**∞](/source/Finiteness_properties_of_groups); (2) All homotopy groups of *F* at infinity are trivial; (3) *F* has no non-abelian free subgroups; (4) *F* is non-amenable. (1) was proved by K. S. Brown and R. Geoghegan in a strong form: there is a K(F,1) with two cells in each positive dimension.[4] (2) was also proved by Brown and Geoghegan [5] in the sense that the cohomology H*(F,ZF) was shown to be trivial; since a previous theorem of M. Mihalik [6] implies that *F* is simply connected at infinity, and the stated result implies that all homology at infinity vanishes, the claim about homotopy groups follows. (3) was proved by M. Brin and C. Squier.[7] The status of (4) is discussed above.

It is unknown if *F* satisfies the [Farrell–Jones conjecture](/source/Farrell%E2%80%93Jones_conjecture). It is even unknown if the Whitehead group of *F* (see [Whitehead torsion](/source/Whitehead_torsion)) or the projective class group of *F* (see [Wall's finiteness obstruction](/source/Wall's_finiteness_obstruction)) is trivial, though it easily shown that *F* satisfies the strong Bass conjecture.

D. Farley [8] has shown that *F* acts as deck transformations on a locally finite CAT(0) [cubical complex](/source/Cubical_complex) (necessarily of infinite dimension). A consequence is that *F* satisfies the [Baum–Connes conjecture](/source/Baum%E2%80%93Connes_conjecture).

## See also

- [Higman group](/source/Higman_group)

- [Non-commutative cryptography](/source/Non-commutative_cryptography)

## References

1. **[^](#cite_ref-1)** Shavgulidze, E. (2009), "The Thompson group F is amenable", *Infinite Dimensional Analysis, Quantum Probability and Related Topics*, **12** (2): 173–191, [doi](/source/Doi_(identifier)):[10.1142/s0219025709003719](https://doi.org/10.1142%2Fs0219025709003719), [MR](/source/MR_(identifier)) [2541392](https://mathscinet.ams.org/mathscinet-getitem?mr=2541392)

1. **[^](#cite_ref-2)** Freyd, Peter; Heller, Alex (1993), "Splitting homotopy idempotents", *[Journal of Pure and Applied Algebra](/source/Journal_of_Pure_and_Applied_Algebra)*, **89** (1–2): 93–106, [doi](/source/Doi_(identifier)):[10.1016/0022-4049(93)90088-b](https://doi.org/10.1016%2F0022-4049%2893%2990088-b), [MR](/source/MR_(identifier)) [1239554](https://mathscinet.ams.org/mathscinet-getitem?mr=1239554)

1. **[^](#cite_ref-3)** Dydak, Jerzy; Minc, Piotr (1977), "A simple proof that pointed FANR-spaces are regular fundamental retracts of ANR's", *Bulletin de l'Académie Polonaise des Science, Série des Sciences Mathématiques, Astronomiques et Physiques*, **25**: 55–62, [MR](/source/MR_(identifier)) [0442918](https://mathscinet.ams.org/mathscinet-getitem?mr=0442918)

1. **[^](#cite_ref-4)** Brown, K.S.; Geoghegan, Ross (1984), *An infinite-dimensional torsion-free FP_infinity group*, vol. 77, pp. 367–381, [Bibcode](/source/Bibcode_(identifier)):[1984InMat..77..367B](https://ui.adsabs.harvard.edu/abs/1984InMat..77..367B), [doi](/source/Doi_(identifier)):[10.1007/bf01388451](https://doi.org/10.1007%2Fbf01388451), [MR](/source/MR_(identifier)) [0752825](https://mathscinet.ams.org/mathscinet-getitem?mr=0752825)

1. **[^](#cite_ref-5)** Brown, K.S.; Geoghegan, Ross (1985), "Cohomology with free coefficients of the fundamental group of a graph of groups", *[Commentarii Mathematici Helvetici](/source/Commentarii_Mathematici_Helvetici)*, **60**: 31–45, [doi](/source/Doi_(identifier)):[10.1007/bf02567398](https://doi.org/10.1007%2Fbf02567398), [MR](/source/MR_(identifier)) [0787660](https://mathscinet.ams.org/mathscinet-getitem?mr=0787660)

1. **[^](#cite_ref-6)** Mihalik, M. (1985), "Ends of groups with the integers as quotient", *Journal of Pure and Applied Algebra*, **35**: 305–320, [doi](/source/Doi_(identifier)):[10.1016/0022-4049(85)90048-9](https://doi.org/10.1016%2F0022-4049%2885%2990048-9), [MR](/source/MR_(identifier)) [0777262](https://mathscinet.ams.org/mathscinet-getitem?mr=0777262)

1. **[^](#cite_ref-7)** Brin, Matthew.; Squier, Craig (1985), "Groups of piecewise linear homeomorphisms of the real line", *[Inventiones Mathematicae](/source/Inventiones_Mathematicae)*, **79** (3): 485–498, [Bibcode](/source/Bibcode_(identifier)):[1985InMat..79..485B](https://ui.adsabs.harvard.edu/abs/1985InMat..79..485B), [doi](/source/Doi_(identifier)):[10.1007/bf01388519](https://doi.org/10.1007%2Fbf01388519), [MR](/source/MR_(identifier)) [0782231](https://mathscinet.ams.org/mathscinet-getitem?mr=0782231)

1. **[^](#cite_ref-8)** Farley, D. (2003), "Finiteness and CAT(0) properties of diagram groups", *[Topology](/source/Topology_(journal))*, **42** (5): 1065–1082, [doi](/source/Doi_(identifier)):[10.1016/s0040-9383(02)00029-0](https://doi.org/10.1016%2Fs0040-9383%2802%2900029-0), [MR](/source/MR_(identifier)) [1978047](https://mathscinet.ams.org/mathscinet-getitem?mr=1978047)

- [Cannon, J. W.](/source/James_W._Cannon); [Floyd, W. J.](/source/William_Floyd_(mathematician)); Parry, W. R. (1996), ["Introductory notes on Richard Thompson's groups"](https://www.imo.universite-paris-saclay.fr/~emmanuel.breuillard/Cannon.pdf) (PDF), *L'Enseignement Mathématique*, IIe Série, **42** (3): 215–256, [ISSN](/source/ISSN_(identifier)) [0013-8584](https://search.worldcat.org/issn/0013-8584), [MR](/source/MR_(identifier)) [1426438](https://mathscinet.ams.org/mathscinet-getitem?mr=1426438)

- Cannon, J.W.; Floyd, W.J. (September 2011). ["WHAT IS...Thompson's Group?"](http://www.ams.org/notices/201108/rtx110801112p.pdf) (PDF). *[Notices of the American Mathematical Society](/source/Notices_of_the_American_Mathematical_Society)*. **58** (8): 1112–1113. [ISSN](/source/ISSN_(identifier)) [0002-9920](https://search.worldcat.org/issn/0002-9920). Retrieved December 27, 2011.

- Geoghegan, Ross (2008), *Topological Methods in Group Theory*, [Graduate Texts in Mathematics](/source/Graduate_Texts_in_Mathematics), vol. 243, [Springer Verlag](/source/Springer_Verlag), [arXiv](/source/ArXiv_(identifier)):[math/0601683](https://arxiv.org/abs/math/0601683), [doi](/source/Doi_(identifier)):[10.1142/S0129167X07004072](https://doi.org/10.1142%2FS0129167X07004072), [ISBN](/source/ISBN_(identifier)) [978-0-387-74611-1](https://en.wikipedia.org/wiki/Special:BookSources/978-0-387-74611-1), [MR](/source/MR_(identifier)) [2325352](https://mathscinet.ams.org/mathscinet-getitem?mr=2325352)

- [Higman, Graham](/source/Graham_Higman) (1974), [*Finitely presented infinite simple groups*](https://books.google.com/books?id=LPvuAAAAMAAJ), Notes on Pure Mathematics, vol. 8, Department of Pure Mathematics, Department of Mathematics, I.A.S. Australian National University, Canberra, [ISBN](/source/ISBN_(identifier)) [978-0-7081-0300-5](https://en.wikipedia.org/wiki/Special:BookSources/978-0-7081-0300-5), [MR](/source/MR_(identifier)) [0376874](https://mathscinet.ams.org/mathscinet-getitem?mr=0376874)

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