1. Thompson’s Group Jim Belk

2. Associative Laws Let  be the following piecewise-linear homeomorphism of  :

3. Associative Laws This homeomorphism corresponds to the operation   . It is called the basic associative law.     

4. Associative Laws Here’s a different associative law . It corresponds to   .

5. Associative Laws           A dyadic subdivision of    is any subdivision obtained by repeatedly cutting intervals in half:

6. Associative Laws An associative law is a PL-homeomorphism that maps linearly between the intervals of two dyadic subdivisions.

7. Associative Laws                  

8. Thompson’s Group Thompson’s Group is the group of all associative laws (under composition).

9. Thompson’s Group Thompson’s Group is the group of all associative laws (under composition). If  , then: Every slope of  is a power of 2. Every breakpoint of  has dyadic rational coordinates. The converse also holds. 2 ½ 1 (¼,½)(¼,½) (½,¾)

10. Properties of is an infinite discrete group.

11. Properties of is an infinite discrete group. is torsion-free.

12. Properties of is an infinite discrete group. is torsion-free. is generated by  and .

13. Properties of is an infinite discrete group. is torsion-free. is generated by  and . is finitely presented (two relations).

14. Properties of is an infinite discrete group. is torsion-free. is generated by  and . is finitely presented (two relations).    is simple. Every proper quotient of is abelian.

15. Geometry of Groups

16. The Geometry of Groups Let  be a group with generating set . The Cayley graph     has: One vertex for each element of . One edge for each pair          Free Group

17.      This makes  into a metric space, which lets us study groups as geometric objects.      Free Group

18. For example, we could investigate the volume growth of balls in .      Free Group

19. For example, we could investigate the volume growth of balls in . Polynomial GrowthExponential Growth      Free Group

20. It’s not too hard to show that Thompson’s group has exponential growth. Polynomial GrowthExponential Growth      Free Group

21. The Geometry of has exponential growth. Every nonabelian subgroup of contains       . does not contain the free group on two elements. Balls in are highly nonconvex (Belk and Bux).

22. Amenability

23. The Isoperimetric Constant Let  be the Cayley graph of a group . If  is a finite subset of , its boundary consists of all edges between  and  .

24. The Isoperimetric Constant Let  be the Cayley graph of a group . The isoperimetric constant is:     is amenable if    .

25. Amenability Example.    is amenable: as   . For an    square,

26. Amenability Example. The free group on two generators is not amenable. In fact:       for any finite subset . So the isoperimetric constant is .

27. Is Amenable? This question has been open for decades. For most groups of interest, the following algorithm determines amenability: 1.Does  contain the free group on two generators? If so, then  is not amenable. 2.Does  have subexponential growth? If so, then  is amenable. 3.Can  be built out of known amenable groups using extensions and unions? If so, then  is amenable. But it doesn’t work on.

28. Some Modest Progress The following is joint work with Ken Brown: 1.We have invented a new way of looking at called “forest diagrams” that simplifies the action of the generators  and . 2.Using forest diagrams, we have derived a formula for the metric on. 3.Using forest diagrams, we have constructed a sequence of (convex) sets in whose isoperimetric ratios approach .