Thompson groups

In mathematics, the Thompson groups (also called Thompson's groups, vagabond groups or chameleon groups) are three groups, commonly denoted $F \subseteq T \subseteq V$, which were introduced by Richard Thompson in some unpublished handwritten notes in 1965 as a possible counterexample to von Neumann conjecture.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:

$\langle A,B \mid\ [AB^{-1},A^{-1}BA] = [AB^{-1},A^{-2}BA^{2}] = \mathrm{id} \rangle$

where [x,y] is the usual group theory commutator, xyx⁻¹y⁻¹.

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

$\langle x_0, x_1, x_2, \dots\ \mid\ x_k^{-1} x_n x_k = x_{n+1}\ \mathrm{for}\ k<n \rangle.$

The two presentations are related by x₀=A, x_n = A¹⁻ⁿBAⁿ⁻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, and as the group of 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. 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

• Higman group
• Non-commutative cryptography

References

1. Shavgulidze, E. (2009), "The Thompson group F is amenable", Inf. Dimen. Anal. Quantum Probab. Relat. Top. 12: 173–191, doi:10.1142/s0219025709003719, MR 2541392
2. Freyd, Peter; Heller, Alex (1993), "Splitting homotopy idempotents", J. Pure Appl. Algebra 89: 93–106, doi:10.1016/0022-4049(93)90088-b, MR 1239554
3. Dydak, Jerzy; Minc, Piotr (1977), "A simple proof that pointed FANR-spaces are regular fundamental retracts of ANR's", Bull. Acad. Polon. Sci. Ser. Sci. Math. Astonom. Phys. 25: 55–62, MR 0442918
4. Brown, K.S.; Geoghegan, Ross (1984), "An infinite-dimensional torsion-free FP_infinity group", Invent. Math. 77: 367–381, doi:10.1007/bf01388451, MR 0752825
5. Brown, K.S.; Geoghegan, Ross (1985), "Cohomology with free coefficients of the fundamental group of a graph of groups", Comment. Math. Helv. 60: 31–45, doi:10.1007/bf02567398, MR 0787660
6. Mihalik, M. (1985), "Ends of groups with the integers as quotient", J. Pure Appl. Algebra 35: 305–320, doi:10.1016/0022-4049(85)90048-9, MR 0777262
7. Brin, Matthew.; Squier, Craig (1985), "Groups of piecewise linear homeomorphisms of the real line", Invent. Math. 79: 485–498, doi:10.1007/bf01388519, MR 0782231
8. Farley, D. (2003), "Finiteness and CAT(0) properties of diagram groups", Topology 42: 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, 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