Conjugacy and Dynamics in Thompson’s Groups Jim Belk (joint with Francesco Matucci)

Conjugacy and Dynamics in

Thompson’s GroupsJim 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.

Properties of • is infinite and torsion-free.

• is finitely presented.

• has an Eilenberg-Maclane complex with two cells in each

dimension (finiteness property ).

• is simple, and .

• has exponential growth, but does not contain a free group.

• may or may not be amenable.

Other Groups is one of three Thompson groups. The other two are:


• Thompson’s group (similar to but on a circle).


• Thompson’s group (similar to but on a circle).

• Thompson’s group (similar to but not continuous).

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¼

0 1½ ¾

The Conjugacy Problem

A solution to the conjugacy problem in is an algorithm which

decides whether given elements are conjugate:

Let be any group.

Higman (1974): Shows that has solvable conjugacy problem.

Mather (1974): Describes conjugacy in Diff .

Brin and Squier (1987): Describe conjugacy in .

Guba and Sapir (1997): Solve the conjugacy problem in .

B and Matucci (2006): Unified solution to the conjugacy

problems in , , and .

A solution to the conjugacy problem in is an algorithm which

decides whether given elements are conjugate:

Strand Diagrams

Strand DiagramsWe represent elements of using strand diagrams:

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¼

0 1½ ¾

Strand Diagrams

A strand diagram is similar to an automaton.

It takes a number (expressed in

binary) as input, and outputs .

Strand Diagrams




Strand Diagrams




Strand Diagrams




Strand Diagrams




Strand Diagrams




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

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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

Strand Diagrams

Strand Diagrams

General Definition:

1. Acyclic directed graph

2. Embedded in the square

3. One source and one sink

4. Other vertices are splits and merges

Theorem. Every strand diagram

represents an element of

Thompson’s group .

Reduction

Type I

Type II

These two moves are called

reductions.

Neither affects the corresponding

piecewise-linear function.

Reduction

Type I

Type II


reductions.



Reduction

Type I

Type II


reductions.



Reduction

Type I

Type II


reductions.



Reduction

Type I

Type II


reductions.



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.







reduced

Conjugacy

MotivationHere’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:

MotivationHere’s a solution to the conjugacy problem in the free group .

Suppose we are given a reduced word:


Motivation


Two elements of are conjugate if and only if they have the

same reduced circle.

Note: Two reduced circles can be compared in linear time (even

though the obvious algorithm is quadratic.)

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 Definition. An annular strand diagram is a directed graph on the

annulus such that:

1. Every vertex is a split or a merge:

2. Every directed loop has positive index around the central hole.

Fact. Every annular strand diagram can be obtained by

“wrapping” a strand diagram.

ReductionsThere are three types of reductions.

Theorem. Every annular strand

diagram is equivalent to a unique

reduced annular strand diagram.

Type I

Type II Type III

Theorem (Guba and Sapir). Two elements of are conjugate if

and only if they have the same reduced annular strand diagram.

The Solution for

Theorem (Guba and Sapir). 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.

Theorem (B and Matucci). The conjugacy problem in has a

linear-time solution.

The Solution for

Conjugacy in and

Thompson’s Group is similar to , but it acts on the circle.

Thompson’s Group

Elements of can be represented by strand diagrams on a cylinder.

Thompson’s Group


Thompson’s Group


Thompson’s Group


Thompson’s Group Wrapping the cylinder around gives a strand diagram on a torus.

Note: Special cutting class in

.

Thompson’s Group Theorem (B and Matucci). Two elements of are conjugate if

and only if they have the same reduced toral strand diagram.

Note: Special cutting class in

.

Thompson’s Group Thompson’s group is similar to , but the functions are not

required to be continuous.

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¼

0 1½ ¾

Thompson’s Group Thompson’s group is similar to , but the functions are not

required to be continuous.

An abstract strand diagram is any

directed, acyclic graph whose vertices

are merges and splits (plus an input

and an output).

Thompson’s Group When you close an abstract strand

diagram, you must keep track of the

cutting cohomology class.

Theorem (B and Matucci). Two

elements of are conjugate if and

only if they have the same reduced

closed abstract strand diagram.

Brin and Squier

Conjugacy in Brin and Squier (1986) gave a dynamical description of conjugacy

in the group .


in the group .

The first step is to break a function

into its components:

Each is a one-bump function.


in the group .

Theorem. are

conjugate if and only if:

1. They have the same number of

components, and

2. is conjugate to for each .


in the group .

Next, Brin and Squier determined

conjugacy for one-bump functions.

Specifically, they found a piecewise-

linear version of the Mather

invariant.

Dynamics of

Dynamics of The relation between conjugacy and dynamics in is more subtle.

Elements of can have both dyadic and

non-dyadic fixed points.


The components of an element are

the portions between the dyadic

fixed points.



The components of an element are

the portions between the dyadic

fixed points.

Theorem. Two elements of are

conjugate if and only if they have

conjugate components.

Connected DiagramsThe main structural

feature of a connected

annular strand diagram

are its directed loops.

Connected DiagramsThe main structural




If the diagram is reduced,

a loop cannot have both

splits and merges.

The main structural






splits and merges.

So there are split loops,

Connected Diagrams

The main structural






splits and merges.

So there are split loops,

and merge loops.

Connected Diagrams

Directed LoopsEach directed loop indicates a fixed point.





Elements of this conjugacy class have an attracting fixed point,

with binary expansion:

Directed LoopsA split loop gives a repelling fixed point:










Any element in this

conjugacy class has

exactly four fixed points:

The material between the

loops determines the three

“bumps”.

For and Directed loops in and are periodic points.

Theorem. Any element of or has a periodic point.

Corollary (Ghys). Every element of has rational rotation number.

Theorem. Any two subgroups of are conjugate.

Theorem. The conjugacy class of a torsion element in is

determined solely by the set of periods.

Documents

Conjugacy and Dynamics in Thompson’s Groups Jim Belk (joint with Francesco Matucci)