Thompson groups seminar, Fall 2009

Reading list

To answer the question "What is a specific group X?" one responds with an isomorphism class. This is somewhat unsatisfactory and one usually wants sample representatives of that class. Thompson groups have many well known representatives. No one representative is "the best" and it is good to know more than one. A presentation might look like a different way to describe a group, but technically it just specifies another representative of the isomorphism class. However, presentations have their different advantages and it is good to know some of them as well.

Cannon, Floyd and Parry

To start with representatives of the isomorphism classes and some presentations, it is best to start with the paper by Cannon, Floyd and Parry. Here is a preprint of their paper. I think it is mostly identical to the published version, but I am not sure. I will refer to the paper as [CFP].

The notation of [CFP] has started to stick. They mention that several older sources use different letters for the groups, and some newer sources still use other letters, but ninety percent of the papers written since [CFP] agree with the names of the groups used in [CFP]. The representatives that are used might be the same, slightly different or very different. Remember that only an isomorphism class is being specified.

The important parts for F are:

  1. The PL function representative for F in Section 1.
  2. The tree diagrams of Section 2 and the relation to the PL functions of Section 1. This includes the normal form of 2.7, and the set of positive elements of 2.8.
  3. The presentations for F in Section 3.
  4. Items 4.1, 4.2, 4.3, 4.4, 4.5 in Section 4 are quite important. Item 4.3 is useful for proving that a candidate representative of the Thompson group F is actually isomorphic to F. One just finds a non-abelian group that satisfies the relations of F. I usually refer to 4.3 as NPNAQ (No Proper Non-Abelian Quotients).
The rest of Section 4 is interesting but less essential at the beginning.

The group T is introduced in Section 5. The material up to 5.3 is a good introduction. From that point the effort is to prove that the presentation above 5.3 is a correct presentation for T. This is a major effort and can be deferred until later.

The group V is handled similarly in Section 6. The effort to the presentation above 6.2 is to describe the group and its elements and find relations that are satisfied. From 6.2 on is the proof that the presentation is correct. Again, this can be deferred.

The derivations of the presentations for T and V include as a major step that the groups are simple. The proof is very involved since it is done for the abstract presentations rather than for the group of homeomorphisms. The reason for this is that once the simplicity of the presentation is shown, it is a triviality that the presenation is the correct presentation for the group of homeomorphisms. Easier proofs of simplicity exist for the groups of homeomorphisms. This will be discussed later.

Section 7 gives one attempt to generalize F to higher dimensions. No one that I know of has worked with this since [CFP] appeared. Generalizing V has proven to be much easier.

Brown's Finiteness Properties paper

The paper Finteness properties of groups by Ken Brown discusses several classes of groups including Thompson's groups. We include only the pages on Thompson's groups. The groups are represented as automorphisms of a certain algebra in Section 4A. They are related to the discussion in [CFP] by the use of tree pairs. They generalize the usual groups by considering trees with more than two descendants of each node.

In Section 4B, the homeomorphisms are brought in which ties up the circle of points of view. The fact that trees with more than 2 children per node are considered in Section 4A results in PL homeomorphisms with slopes other than powers of 2 being considered here.

Generators and relations are discussed in Section 4C.

The discussion in Section 4D starts an analysis that ends with the proof that certain subgroups are simple. This is a long involved section since many different groups are being considered. It can be deferred until later.

Section 4E gets to the point of the paper: the groups have strong finiteness properties. This refers to parts of the paper that have not been copied. What might be interesting is the action that is defined on related complexes. This section can also be deferred.

Brin-Guzman

Section 2 of the Brin-Guzman paper discusses some of the groups in Browns' finiteness properties paper and some generalizations. The representation here is of PL homeomorphisms on the half line. The fact that the line and half line can be used as successfully as the unit interval is useful. There is minor reference to Section 1 in that the groups are put into two large classes (A and B). Because of this some of the material in Section 1 (through Section 1.4) would be good to read.

In Section 2, there is much overlap with Brown's paper and should be easy going. The material in 2.5 does not appear in Brown (it appears in other places) and should be read. Nothing after Section 2.5 is worth reading at this point. The proofs of simplicity are less involved than the proofs in [CFP] or in Brown's finiteness properties paper. They are less involved than the proofs in [CFP] since they are about the homeomorphism groups and not the presentations and are less involved than in the Brown paper since fewer groups are considered. The argument of Higman in 2.4.2(g) is very important. It is repeated with less machinery surrounding it in the paper below on higher dimensional groups.

Higher dimensional groups

The paper on higher dimensional groups has lots of nice pictures relating to a generalization of V (the evidence that V is easier to move into higher dimensions than F) and two proofs of simplicity. One (for [2V, 2V]) is geometric (3.3 which is based on the more painful 3.2) and one (for V) is very combinatorial and is almost trivial (Section 12) and is included to show how easy it is to prove simplicity for V and how much harder it apparently is to prove it for 2V. It turns out to be almost as easy to prove for 2V and the needed ingredients are given here.

Brown-Geoghegan

The paper by Brown and Geoghegan has a short introduction to the group F. It includes discussions of certain points (normal form, free abelian subgroups, universal properties) but assumes some results such as NPNAQ. We only include the relevant pages.

Thompson's notes

You will see references in various papers to "widely circulated, handwritten notes of Richard J. Thompson." They are here scanned in two groups. They are scanned at high resolution since they are hard to read and so they are in two files: Pages 1-7 and Pages 8-11. They are more readable than they look at first, but are rough going. They are not that continuous and take huge jumps in spots. The ending pages are rather scattered.