Thompson’s Group 
Jim Belk
Associative Laws
Let  be the following piecewise-linear
homeomorphism of :
Associative Laws
This homeomorphism corresponds to the
operation   .
It is called the basic associative law.






Associative Laws
Here’s a different associative law .
It corresponds to   .
Associative Laws
A dyadic subdivision of   is any subdivision
obtained by repeatedly cutting intervals in
half:












Associative Laws
An associative law is a PL-homeomorphism that
maps linearly between the intervals of two
dyadic subdivisions.
Associative Laws
   





 





Thompson’s Group 
Thompson’s Group  is the group of all
associative laws (under composition).
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
Properties of 
•
 is an infinite discrete group.
Properties of 
•
 is an infinite discrete group.
•
 is torsion-free.
Properties of 
•
 is an infinite discrete group.
•
 is torsion-free.
•
 is generated by  and .
Properties of 
•
 is an infinite discrete group.
•
 is torsion-free.
•
 is generated by  and .
•
 is finitely presented (two relations).
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.
Geometry of
Groups
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
This makes  into a metric space, which lets
us study groups as geometric objects.

Free Group
For example, we could investigate the volume
growth of balls in .

Free Group
For example, we could investigate the volume
growth of balls in .
Polynomial Growth
Exponential Growth

Free Group
It’s not too hard to show that Thompson’s
group  has exponential growth.
Polynomial Growth
Exponential Growth

Free Group
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).
Amenability
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 .
The Isoperimetric Constant
Let  be the Cayley graph of a group .
The isoperimetric
constant is:
 
 is amenable if   .
Amenability
Example.    is amenable:
For an    square,
as   .
Amenability
Example. The free group on two generators
is not amenable.
In fact:
  
for any finite subset .
So the isoperimetric
constant is .
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 .
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 .