Conjugacy in Thompson’s Group Jim Belk (joint with Francesco Matucci)

Thompson’s Group Thompson’s Group  . 2 ½ 1 (¼,½)(¼,½) (½,¾)   Piecewise-linear homeomorphisms of  .   if and only if: 1. The slopes of  are powers of 2, and 2. The breakpoints of  have dyadic rational coordinates.

Thompson’s Group  . 2 ½ 1 (¼,½)(¼,½) (½,¾)   if and only if: 1. The slopes of  are powers of 2, and 2. The breakpoints of  have dyadic rational coordinates. 10 ½              ¼ 01 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

We represent elements of using strand diagrams: 1 0 ½              ¼ 01 ½ ¾

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 1 0 ½              ¼ 01 ½ ¾

Strand Diagrams 1 0 ½              ¼ 01 ½ ¾     

Strand Diagrams 1 0 ½              ¼ 01 ½ ¾         

Strand Diagrams 1 0 ½              ¼ 01 ½ ¾       

Strand Diagrams

Reduction Type I Type II These two moves are called reductions. Neither affects the corresponding piecewise-linear function.

Reduction Type I Type II These two moves are called reductions. Neither affects the corresponding piecewise-linear function.   

Reduction Type I Type II These two moves are called reductions. Neither affects the corresponding piecewise-linear function.   

Reduction Type I Type II These two moves are called reductions. Neither affects the corresponding piecewise-linear function.    

Reduction Type I Type II These two moves are called reductions. Neither affects the corresponding piecewise-linear function.    

Reduction Type I Type II A strand diagram is reduced if it is not subject to any reductions. Theorem. There is a one-to-one correspondence: 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 A solution to the conjugacy problem in  is an algorithm which decides whether given elements     are conjugate:     Let  be any group. 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.