# Thompson groups

In mathematics, the **Thompson groups** (also called **Thompson's groups**, **vagabond groups** or **chameleon groups**) are three groups, commonly denoted , which were introduced by Richard Thompson in some unpublished handwritten notes in 1965 as a possible counterexample to the von Neumann conjecture.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 which have made them counterexamples to many general conjectures in group theory. All three Thompson groups are infinite but finitely presented. The groups *T* and *V* are (rare) examples of infinite but finitely-presented simple groups. The group *F* is not simple but its derived subgroup [*F*,*F*] is and the quotient of *F* by its derived subgroup is the free abelian group of rank 2. *F* is totally ordered, has exponential growth, and does not contain a subgroup isomorphic to the free group of rank 2.

It is conjectured that *F* is not amenable and hence a further counterexample to the long-standing but recently disproved
von Neumann conjecture for finitely-presented groups: it is known that *F* is not elementary amenable.

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

## Presentations

A finite presentation of *F* is given by the following expression:

where [*x*,*y*] is the usual group theory commutator, *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:

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

## Other representations

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

The group *F* can also be considered as acting on the 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 on one generator.

## Amenability

The conjecture of Thompson that *F* is not amenable 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 review.

It is known that *F* is not elementary amenable, see Theorem 4.10 in Cannon-Floyd-Parry. If *F* is **not** amenable, then it would be another counterexample to the long-standing but recently disproved von Neumann conjecture for finitely-presented groups, which suggested that a finitely-presented group is amenable 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 which was only published much later but was in circulation as a preprint at that time, P. Freyd and A. Heller [2] showed that the *shift map* on *F* induces an unsplittable homotopy idempotent on the Eilenberg-MacLane 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**_{∞}; (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. It is even unknown if the Whitehead group of *F* (see Whitehead torsion) or the projective class group of *F* (see 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 (necessarily of infinite dimension). A consequence is that *F* satisfies the Baum-Connes conjecture.

## See also

## References

- Shavgulidze, E. (2009), "The Thompson group F is amenable",
*Infinite Dimensional Analysis, Quantum Probability and Related Topics*,**12**(2): 173–191, doi:10.1142/s0219025709003719, MR 2541392 - Freyd, Peter; Heller, Alex (1993), "Splitting homotopy idempotents",
*Journal of Pure and Applied Algebra*,**89**(1–2): 93–106, doi:10.1016/0022-4049(93)90088-b, MR 1239554 - 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 0442918 - Brown, K.S.; Geoghegan, Ross (1984),
*An infinite-dimensional torsion-free FP_infinity group*,**77**, pp. 367–381, Bibcode:1984InMat..77..367B, doi:10.1007/bf01388451, MR 0752825 - Brown, K.S.; Geoghegan, Ross (1985), "Cohomology with free coefficients of the fundamental group of a graph of groups",
*Commentarii Mathematici Helvetici*,**60**: 31–45, doi:10.1007/bf02567398, MR 0787660 - Mihalik, M. (1985), "Ends of groups with the integers as quotient",
*Journal of Pure and Applied Algebra*,**35**: 305–320, doi:10.1016/0022-4049(85)90048-9, MR 0777262 - Brin, Matthew.; Squier, Craig (1985), "Groups of piecewise linear homeomorphisms of the real line",
*Inventiones Mathematicae*,**79**(3): 485–498, Bibcode:1985InMat..79..485B, doi:10.1007/bf01388519, MR 0782231 - Farley, D. (2003), "Finiteness and CAT(0) properties of diagram groups",
*Topology*,**42**(5): 1065–1082, doi:10.1016/s0040-9383(02)00029-0, MR 1978047

- Cannon, J. W.; Floyd, W. J.; Parry, W. R. (1996), "Introductory notes on Richard Thompson's groups" (PDF),
*L'Enseignement Mathématique*, IIe Série,**42**(3): 215–256, ISSN 0013-8584, MR 1426438 - Cannon, J.W.; Floyd, W.J. (September 2011). "WHAT IS...Thompson's Group?" (PDF).
*Notices of the American Mathematical Society*.**58**(8): 1112–1113. ISSN 0002-9920. Retrieved December 27, 2011. - Geoghegan, Ross (2008),
*Topological Methods in Group Theory*, Graduate Texts in Mathematics,**243**, Springer Verlag, arXiv:math/0601683, doi:10.1142/S0129167X07004072, ISBN 978-0-387-74611-1, MR 2325352 - Higman, Graham (1974),
*Finitely presented infinite simple groups*, Notes on Pure Mathematics,**8**, Department of Pure Mathematics, Department of Mathematics, I.A.S. Australian National University, Canberra, ISBN 978-0-7081-0300-5, MR 0376874