A Thompson Group
for the Basilica

Jim Belk

A Thompson Group
for the Basilica

Jim Belk

(joint with Bradley Forrest)

Thompson’s Groups

In the 1960’s, Richard J. Thompson defined three groups:

Definition of F

A dyadic subdivision of [0,1] is any subdivision obtained by repeatedly cutting intervals in half:

Definition of F

A dyadic subdivision of [0,1] is any subdivision obtained by repeatedly cutting intervals in half:

Definition of F

A dyadic subdivision of [0,1] is any subdivision obtained by repeatedly cutting intervals in half:

Definition of F

A dyadic subdivision of [0,1] is any subdivision obtained by repeatedly cutting intervals in half:

Definition of F

A dyadic subdivision of [0,1] is any subdivision obtained by repeatedly cutting intervals in half:

The intervals of such a subdivision are standard dyadic intervals:

Definition of F

A dyadic rearrangement of [0,1] is a PL homeomorphism that maps linearly between the intervals of two dyadic subdivisions:

The set of all dyadic rearrangements is Thompson’s group F.

Definition of F

The set of all dyadic rearrangements is Thompson’s group F.

Given an element of F:
    •  Each segment has slope (2^m).
    •  Each breakpoint has dyadic rational coordinates.

These conditions characterize the elements of F.

Properties of F

Properties of F

•  F is finitely generated.

Properties of F

•  F is finitely generated.
•  F is finitely presented.

Properties of F

•  F is finitely generated.
•  F is finitely presented.

•  [ F, F ] is simple, and F= [ F, F ] is isomorphic to Z × Z.

Definition of T

We can also define dyadic rarrangements for the circle:

Definition of T

We can also define dyadic rarrangements for the circle:

The group of all such rearrangements is Thompson’s group T.

Properties of T

Properties of T

•  T is finitely generated.

Properties of T

•  T is finitely generated.
•  T is finitely presented.
•  T is a simple group.

Definition of V

Thompson’s group V acts on the Cantor set.

Definition of V

Thompson’s group V acts on the Cantor set.

Definition of V

Thompson’s group V acts on the Cantor set.

Definition of V

Thompson’s group V acts on the Cantor set.

Properties:
      •  V is finitely generated.
      •  V is finitely presented.
      •  V is a simple group.

Also, F ½ T ½ V.

Generalizations

Basic Questions:
      Why are there three Thompson groups?
      What do the interval, circle, and Cantor set have in common?

Generalizations

Basic Questions:
      Why are there three Thompson groups?
      What do the interval, circle, and Cantor set have in common?

Generalizations:
      •  F(n), T(n), and V(n)  (Higman 1974, Brown 1987)
      •  Other PL groups  (Bieri & Strebel 1985, Stein 1992)
      •  Diagram Groups  (Guba & Sapir 1997)
      •  “Braided” V  (Brin 2004, Dehornoy 2006)
      •   2V  (Brin 2004)

Self-Similarity

Self-Similarity

The definition of F depends on the self-similarity of the interval.

Self-Similarity

The definition of F depends on the self-similarity of the interval.

Self-Similarity

This concept is borrowed from fractal geometry.

Self-Similarity

Some fractals have the same self-similarity structure as an interval:

Self-Similarity

Some fractals have the same self-similarity structure as an interval:

Self-Similarity

Some fractals have the same self-similarity structure as an interval:

Thompson’s group F acts on such a fractal by piecewise-similarities.

Self-Similarity

Thompson’s group F acts on such a fractal by piecewise similarities.

Self-Similarity

Thompson’s group F acts on such a fractal by piecewise similarities.

There are also fractals on which T or V acts by piecewise similarities.

Self-Similarity

Thompson’s group F acts on such a fractal by piecewise similarities.

There are also fractals on which T or V acts by piecewise similarities.

Perhaps we can find new Thompson groups associated to other fractal shapes.

Julia Sets

Julia Sets

Let  f(z)=(z^2)+c, where z and c are complex.

The filled Julia set for  f  is the set

Example:  The filled Julia set for  f(z)=(z^2) is the closed unit disk.

Orbits Under (z^2)

Orbits Under (z^2)-1

Orbits Under (z^2)-1

Filled Julia Set for (z^2)-1

Filled Julia Set for (z^2)-1

This Julia set is known as the Basilica.

Filled Julia Set

Julia Set

Self-Similarity

The Julia set has a self-similar structure:

Note:  The “similarity” maps are not actually linear.

Self-Similarity

External Angles

External Angles

The complement of the Basilica is simply connected in the Riemann sphere:

External Angles

The complement of a disk is also simply connected:

External Angles

So there exists a Riemann map between the two complements.

External Angles

So there exists a Riemann map between the two complements.

External Angles

The radial lines are called external rays.

External Angles

These define the external angles.

External Angles

Pinch Points:  Denominator is 3 · (2^n)

External Angles

This picture is called the invariant lamination for the Basilica.

External Angles

Homeomorphisms that preserve the lamination descend to B.

Transformations
of the Basilica

Transformations

We can define piecewise-linear tranformations of the Basilica.

Transformations

We can define piecewise-linear tranformations of the Basilica.

f(θ)=(Piecewise[{{θ/2, if -1/3≤θ≤1/3}, {2θ - 1/3, if 1/3≤θ≤2/3}}])

Transformations

We can define piecewise-linear tranformations of the Basilica.

f(θ)=(Piecewise[{{θ/2, if -1/3≤θ≤1/3}, {2θ - 1/3, if 1/3≤θ≤2/3}}])

Transformations

Transformations

Transformations

The Group (Subscript[T, B])

Definition.  Let (Subscript[T, B]) be the group of all piecewise-linear homeomorphisms of the circle satisfying the following conditions:
    •  The homeomorphism preserves the invariant lamination.
    •  Every breakpoint maps a pinch point (denominator 3 · (2^n))
           to a pinch point.

Theorem (Belk, Forrest).  If g ∈ (Subscript[T, B]), then the slope of each segment of g is a power of 2.

Theorem (Belk, Forrest).  The group (Subscript[T, B]) contains an isomorphic copy of Thompson’s group T.

Theorem (Belk, Forrest).  The group (Subscript[T, B]) contains an isomorphic copy of Thompson’s group T.

Theorem (Belk, Forrest).  The group (Subscript[T, B]) contains an isomorphic copy of Thompson’s group T.

Theorem (Belk, Forrest).  Thompson’s group T contains an isomorphic copy of (Subscript[T, B]).

Theorem (Belk, Forrest).  The group (Subscript[T, B]) contains an isomorphic copy of Thompson’s group T.

Theorem (Belk, Forrest).  Thompson’s group T contains an isomorphic copy of (Subscript[T, B]).

Argument:  The group of all PL homeomorphisms with slopes (2^n)
                    and breakpoints with denominators (2^n) or 3 · (2^n) is
                    isomorphic to T.

Theorem (Belk, Forrest).  (Subscript[T, B]) is generated by four elements.

The generators are:
    •  Three generators for the copy of T.
    •  The horizontal shift:

Theorem (Belk, Forrest).  (Subscript[T, B]) is generated by four elements.

The generators are:
    •  Three generators for the copy of T.
    •  The horizontal shift.

Note:  We suspect that (Subscript[T, B]) is not finitely presented.

Theorem (Belk, Forrest).  The commutator subgroup of (Subscript[T, B]) has index two.  It consists of all elements that preserve the following coloring:

Theorem (Belk, Forrest).  The commutator subgroup is simple.

Other Julia Sets

Other Julia Sets

Each quadratic polynomial  f(z)=(z^2)+c gives a Julia set.

Other Julia Sets

The Mandelbrot set describes the possible values of c.

Other Julia Sets

The definition of (Subscript[T, B]) should generalize easily to the family of rabbits.

Other Julia Sets

The definition of (Subscript[T, B]) should generalize easily to the family of rabbits.

Other Julia Sets

Other Julia sets may be more difficult.

Other Julia Sets

Other Julia sets may be more difficult.

Other Julia Sets

Can we find a uniform definition?

The End