Thompson groups

This page is about the infinite Thompson groups F, T and V. For the sporadic finite simple group Th see Thompson sporadic group.

In mathematics, the Thompson groups (also called Thompson's groups, vagabond groups or chameleon groups) are three groups, commonly denoted F⊂T⊂V, which were introduced by Richard Thompson in some unpublished handwritten notes in 1965. 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 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.

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).

See also

• Higman group

References

• Cannon, J. W.; Floyd, W. J.; Parry, W. R. (1996), "Introductory notes on Richard Thompson's groups" (PDF), L'Enseignement Mathématique. Revue Internationale. IIe Série 42 (3): 215–256, ISSN 0013-8584, MR1426438, http://www.math.binghamton.edu/matt/thompson/cfp.pdf 
• 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. http://www.ams.org/notices/201108/rtx110801112p.pdf. Retrieved December 27, 2011. 
• 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, MR0376874, http://books.google.com/books?id=LPvuAAAAMAAJ