Exploring Thompson’s Group: Associative Laws and Group Properties

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. ½ 1 (½,¾) (¼,½) 2

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 ofGroups

16.   Free Group 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 

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

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

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

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

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 constantis:    is amenable if   .

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

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: • Does  contain the free group on two generators? If so, then  is not amenable. • Does  have subexponential growth? If so, then  is amenable. • 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: • We have invented a new way of looking at  called “forest diagrams” that simplifies the action of the generators  and . • Using forest diagrams, we have derived a formula for the metric on . • Using forest diagrams, we have constructed a sequence of (convex) sets in  whose isoperimetric ratios approach .