Conjugacy inThompson’s Group

Jim Belk

(joint with Francesco Matucci)

Thompson’s Group

Thompson’s Group .

2

½

1

(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)

(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

0 1½ 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 DiagramsWe represent elements of using strand diagrams:

10 ½

¼

0 1½ 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

10 ½

¼

0 1½ ¾

Strand Diagrams

10 ½

¼

0 1½ ¾

Strand Diagrams

10 ½

¼

0 1½ ¾

Strand Diagrams

10 ½

¼

0 1½ ¾

Strand Diagrams

Strand Diagrams

Strand Diagrams

Strand Diagrams

Strand Diagrams

Strand Diagrams

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:

reducedstrand

diagrams

elements of

MultiplicationWe can multiply two strand diagrams concatenating them:

MultiplicationUsually the result will not be reduced.

MultiplicationUsually the result will not be reduced.

MultiplicationUsually 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 GroupHere’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 ResultTheorem. Two elements of are conjugate if and only if they

have the same reduced annular strand diagram.

Main ResultTheorem (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 .