Conjugacy in
Thompson’s Group 
Jim Belk
(joint with Francesco Matucci)
Thompson’s Group 
  Piecewise-linear homeomorphisms of  .
Thompson’s Group   .
½
   if and only if:
1
1. The slopes of  are powers of 2, and
(¼,½)
2. The breakpoints of  have dyadic
rational coordinates.
2
(½,¾)
Thompson’s Group   .
½
   if and only if:
1
1. The slopes of½ are powers of 2, and
0
¼
1
2. The breakpoints of  have dyadic
rational coordinates.
(¼,½)

 
2
  

0
½
¾
1
(½,¾)
 
Another Example
  
  
  
  


 
 
Another Example
  
  
  
  


 
 
Another Example
  
  
  
  


 
 
Another Example
  
  
  
  


 
 
Another Example
  
  
  
  


 
 
Another Example
  
  
  
  


 
 
Another Example
  
  
  
  


 
 
Another Example
In general, a dyadic subdivision is any subdivision of 
obtained by repeatedly cutting intervals in half.
Every element of  maps linearly between the intervals of two
dyadic subdivisions.
Strand Diagrams
Strand Diagrams
We represent elements of  using strand diagrams:
0
¼
0
½
½
1
¾
  
  

 
1
Strand Diagrams


A strand diagram takes a number    
(expressed in binary) as input, and
outputs .
Strand Diagrams

A strand diagram takes a number    
(expressed in binary) as input, and
outputs .
Strand Diagrams


A strand diagram takes a number    
(expressed in binary) as input, and
outputs .
Strand Diagrams



A strand diagram takes a number    
(expressed in binary) as input, and
outputs .
Strand Diagrams




A strand diagram takes a number    
(expressed in binary) as input, and
outputs .
Strand Diagrams





A strand diagram takes a number    
(expressed in binary) as input, and
outputs .
Strand Diagrams
Every vertex (other than the top and the
bottom) is either a split or a merge:
split
merge
Strand Diagrams
A split removes the first digit of a binary
expansion:


merge


Strand Diagrams
A merge inserts a new digit:







Strand Diagrams
0
¼
0
½
½
1
¾
  
  

 
1
Strand Diagrams

0
¼
½
1

0
½
¾
  

  


 
1
Strand Diagrams

0
¼
½
1


0


½
¾
  
  

 
1
Strand Diagrams

0
¼
½
1

0


½
¾
  
  

 
1
Strand Diagrams
Strand Diagrams
Strand Diagrams
Strand Diagrams
Strand Diagrams
Strand Diagrams
Strand Diagrams
Reduction
Type I
These two moves are called
reductions.
Neither affects the corresponding
piecewise-linear function.
Type II
Reduction

Type I
These two moves are called
reductions.

Neither affects the corresponding
piecewise-linear function.

Type II
Reduction

Type I

These two moves are called
reductions.
Neither affects the corresponding
piecewise-linear function.

Type II
Reduction
Type I
These two moves are called
reductions.
Neither affects the corresponding
piecewise-linear function.

Type II


Reduction
Type I
These two moves are called
reductions.
Neither affects the corresponding
piecewise-linear function.

Type II


Reduction
Type I
A strand diagram is reduced
if it is not subject to any
reductions.
Theorem. There is a one-to-one
correspondence:
Type II
reduced
strand
diagrams
elements
of 
Multiplication
We can multiply two strand diagrams concatenating them:



Multiplication
Usually the result will not be reduced.



Multiplication
Usually the result will not be reduced.



Multiplication
Usually the result will not be reduced.



Conjugacy
The Conjugacy Problem
Let  be any group.
A solution to the conjugacy problem in  is an algorithm which
decides whether given elements     are conjugate:
  
Classical Algorithm Problems:
• Word Problem
• Conjugacy Problem
• Isomorphism Problem
The Free Group
Here’s a solution to the conjugacy problem in the free group  .
Suppose we are given a reduced word:
       
To find the conjugacy class, make the word into a circle and
reduce:














The Free Group
To find the conjugacy class, make the word into a circle and
reduce:






Two elements of   are conjugate if and
 only if they
 have the


same
reduced circle.





The Solution for 
The idea is to wrap the strand diagram around in a circle:
We call this an annular strand diagram.
The Solution for 
The idea is to wrap the strand diagram around in a circle:
We call this an annular strand diagram.
The Solution for 
The idea is to wrap the strand diagram around in a circle:
We call this an annular strand diagram.
The Solution for 
The idea is to wrap the strand diagram around in a circle:
We call this an annular strand diagram.
The Solution for 
The idea is to wrap the strand diagram around in a circle:
We call this an annular strand diagram.
The Solution for 
The idea is to wrap the strand diagram around in a circle:
We call this an annular strand diagram.
The Solution for 
The idea is to wrap the strand diagram around in a circle:
We call this an annular strand diagram.
The Solution for 
The idea is to wrap the strand diagram around in a circle:
We call this an annular strand diagram.
Main Result
Theorem. Two elements of  are conjugate if and only if they
have the same reduced annular strand diagram.
Main Result
Theorem (B and Matucci). Two elements of  are conjugate if
and only if they have the same reduced annular strand diagram.
Hopcroft and Wong (1974): You can determine whether two
planar graphs are isomorphic in linear time.
Corollary. The conjugacy problem in  has a linear-time solution.
By analyzing the structure of the annular strand diagram, one can
get a complete description of the dynamics of an element of .