Rearrangement Groups
of Fractals

Jim Belk
Bard College

(joint with Bradley Forrest)

Thompson’s Groups

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

Definition of F

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

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

Definitions of T and V

T is the group of dyadic rearrangements of a circle.

Definitions of T and V

T is the group of dyadic rearrangements of a circle.

V is the group of dyadic rearrangements of a Cantor set.

Definitions of T and V

T is the group of dyadic rearrangements of a circle.

V is the group of dyadic rearrangements of a Cantor set.

Definitions of T and V

T is the group of dyadic rearrangements of a circle.

V is the group of dyadic rearrangements of a Cantor set.

Properties of F, T, and V

•  F embeds into T, and T embeds into V.

•  F, T, and V are finitely presented.

•  T and V are simple, as is [ F, F ].

•  (Brown & Geoghegan) F, T, and V have type (Subscript[F, ∞]).

•  (Farley) F, T, and V act properly and isometrically on CAT(0) cubical complexes.

Generalizations

Basic Questions:
      Why are there three Thompson groups?
      What’s special about the interval, circle, and Cantor set?

Generalizations

Basic Questions:
      Why are there three Thompson groups?
      What’s special about the interval, circle, and Cantor set?

Generalizations:
      •  F(n), T(n), 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)

Main Idea

The definitions of F, T, and V depend on the self-similarity of the interval and the Cantor set.

Main Idea

Maybe we can find Thompson-like groups associated with other self-similar spaces.

The Basilica

The Basilica

The following fractal is called the Basilica.

It’s the Julia set for the polynomial f(z) = (z^2)-1.

Bulbs and Edges

These structures are all homeomorphic.  Let’s call them bulbs.

Bulbs and Edges

The Basilica is actually the union of two bulbs.

Bulbs and Edges

Each bulb can be partitioned into one bulb and two edges.

Bulbs and Edges

Each bulb can be partitioned into one bulb and two edges.

Bulbs and Edges

Similarly, each edge can be partitioned into one bulb and two edges.

Bulbs and Edges

Similarly, each edge can be partitioned into one bulb and two edges.

Two Subdivision Moves

Allowed Subdivisions

An allowed subdivision of the Basilica is obtained by repeatedly applying these two moves, starting with the two main bulbs.

Allowed Subdivisions

An allowed subdivision of the Basilica is obtained by repeatedly applying these two moves.

Starting Subdivision

Allowed Subdivisions

An allowed subdivision of the Basilica is obtained by repeatedly applying these two moves.

Allowed Subdivisions

An allowed subdivision of the Basilica is obtained by repeatedly applying these two moves.

The Basilica Group

A rearrangement of the Basilica is a homeomorphism that maps conformally between the pieces of two allowed subdivisions.

The group of all rearrangements is the Basilica Thompson group (Subscript[T, B]).

Example Rearrangement

Example Rearrangement

Properties of (Subscript[T, B])

Theorem (B-F)
  1.  (Subscript[T, B])  is finitely generated. 
  2.  [ (Subscript[T, B]), (Subscript[T, B]) ]  has index two and is simple. 
  3.  (Subscript[T, B]) embeds in T, and T embeds in (Subscript[T, B]).

Conjecture
(Subscript[T, B]) is not finitely presented.

Edge Replacement
Systems

Edge Replacement Systems

An edge replacement system consists of the following data:

Edge Replacement Systems

An edge replacement system consists of the following data:

1.  A replacement rule e → R.

R is a directed graph with two distinguished vertices.

Edge Replacement Systems

An edge replacement system consists of the following data:

1.  A replacement rule e → R.

2.  A base graph (Subscript[G, 0]).

Edge Replacement Systems

An expansion of (Subscript[G, 0]) is any graph obtained by repeatedly applying the replacement rule.

Edge Replacement Systems

An expansion of (Subscript[G, 0]) is any graph obtained by repeatedly applying the replacement rule.

Edge Replacement Systems

An expansion of (Subscript[G, 0]) is any graph obtained by repeatedly applying the replacement rule.

Edge Replacement Systems

An expansion of (Subscript[G, 0]) is any graph obtained by repeatedly applying the replacement rule.

Edge Replacement Systems

An expansion of (Subscript[G, 0]) is any graph obtained by repeatedly applying the replacement rule.

Edge Replacement Systems

A full expansion consists of applying the rule to every edge exactly once.

Edge Replacement Systems

A full expansion consists of applying the rule to every edge exactly once.

Edge Replacement Systems

A full expansion consists of applying the rule to every edge exactly once.

Iterating this procedure gives a sequence of full expansions.

Edge Replacement Systems

A full expansion consists of applying the rule to every edge exactly once.

Iterating this procedure gives a sequence of full expansions.

Edge Replacement Systems

A full expansion consists of applying the rule to every edge exactly once.

Iterating this procedure gives a sequence of full expansions.

Edge Replacement Systems

A full expansion consists of applying the rule to every edge exactly once.

Iterating this procedure gives a sequence of full expansions.

Edge Replacement Systems

A full expansion consists of applying the rule to every edge exactly once.

Iterating this procedure gives a sequence of full expansions.

The Limit Space

Given an edge replacement system:

The sequence of full expansions converges to a limit space:

The Limit Space

Theorem
Suppose the replacement graph R is expanding, i.e.
  1.  The initial and terminal vertices are not adjacent in R. 
  2.  R has at least three vertices, and no isolated vertices.
Then the limit space X is compact and metrizable.

Rearrangements

Each expansion of (Subscript[G, 0]) corresponds to a subdivision of the limit space.

Rearrangements

An isomorphism between two expansions defines a homeomorphism of the limit space.

This is called a rearrangement.

Rearrangements

Theorem
The rearrangements of the limit space form a group.

This is the rearrangement group of the limit space (w.r.t. the given edge replacement system).

Example: Thompson’s Groups

Example: Thompson’s Groups

The Douady Rabbit

The limit space is a Julia set known as the “Douady rabbit”.

A Rational Julia Set

The limit space is homeomorphic to the Julia set for (z^(-2))-1.

Example:
The Vicsek Fractal

The Vicsek Fractal

The Vicsek Fractal is a certain fractal tree in the plane:

The Vicsek Fractal

The Vicsek Fractal is a certain fractal tree in the plane:

The Vicsek Fractal

The Vicsek Fractal is a certain fractal tree in the plane:

The Vicsek Fractal

The Vicsek Fractal is a certain fractal tree in the plane:

The Vicsek Fractal

The Vicsek Fractal is a certain fractal tree in the plane:

The Vicsek Fractal

The Vicsek Fractal is a certain fractal tree in the plane:

The Vicsek Fractal

The Vicsek Fractal is a certain fractal tree in the plane:

The Vicsek Fractal

The Vicsek Fractal is a certain fractal tree in the plane:

The Vicsek Fractal

The Vicsek Fractal is a certain fractal tree in the plane:

The Vicsek Fractal

The Vicsek Fractal is a certain fractal tree in the plane:

The Vicsek Fractal

The following edge replacement system produces this fractal:

(Row[(Row), ])

(Row[(Row), ])

(Row[(Row), ])

(Row[(Row), ])

(Row[(Row), ])

The Vicsek Fractal

Here is an example rearrangement:

The Vicsek Fractal

Theorem (B-F)
Let (Subscript[T, Vic]) be the space rearrangement group of the Vicsek fractal.
  1.  (Subscript[T, Vic]) contains copies of Thompson’s group F.
  2.  (Subscript[T, Vic]) is contained in Thompson’s group V.
  3.  (Subscript[T, Vic]) has type (Subscript[F, ∞]).

Interlude:
Finiteness Properties

Finiteness Properties

Let G be a group.

•  G has type (Subscript[F, 1]) if G is finitely generated.
    •  G has type (Subscript[F, 2]) if G is finitely presented.
                   ⋮
    •  G has type (Subscript[F, n]) if there exists a K(G,1) with finite n-skeleton.
                   ⋮
    •  G has type (Subscript[F, ∞]) if there exists a K(G,1) with finitely many cells of each dimension.

The Thompson groups F, T, and V all have type (Subscript[F, ∞]).

Some Technology

There is a “standard” procedure for proving that a group has type (Subscript[F, ∞]) (due to Brown, Bestvina, Brady, etc.):

1.  Find a nice action of the group on a complex K. 
    2.  Choose an appropriate Morse function φ: K -> [0, ∞). 
    3.  Analyze the connectivity of the descending links.

Some Technology

There is a “standard” procedure for proving that a group has type (Subscript[F, ∞]) (due to Brown, Bestvina, Brady, etc.):

1.  Find a nice action of the group on a complex K. 
    2.  Choose an appropriate Morse function φ: K -> [0, ∞). 
    3.  Analyze the connectivity of the descending links.

In our case, K will be a CAT(0) cubical complex, so the descending links will be flag simplicial complexes.

The Cubical Complex

The Cubical Complex

Given a replacement system:

We wish to define a CAT(0) cubical complex on which the resulting group acts.

We start by defining a groupoid of rearrangements.

Given an edge replacement rule:

Each base graph G has a corresponding limit space X(G):

Given an edge replacement rule:

Each base graph G has a corresponding limit space X(G):

A Groupoid of Rearrangements

It makes sense to talk about rearrangements between two such limit spaces.

For a given replacement rule, the set of all such rearrangements forms a groupoid.

A Groupoid of Rearrangements

Example
Suppose G' is obtained from G by replacing an edge. Then the canonical rearrangement X(G)->X(G') is called a split.

(Row)

Fact
The splits generate the groupoid of rearrangements.

Split Squares

Splits along two different edges form a commutative square:

The Cubical Complex

Given a replacement system:

The complex K is defined as follows:
    •  One vertex for each rearrangement X((Subscript[G, 0]))->X(G)
            (up to isomorphism of G).
    •  Edges in K correspond to composing with a split.

The Cubical Complex

Theorem (B-F)
Given an expanding replacement rule e -> R and base graph (Subscript[G, 0]): 
  1.  The resulting complex K is a CAT(0) cubical complex.
  2.  The rearrangement group acts on K by isometries.
  3.  Vertex stabilizers for this action are finite.

In the case of F, T, or V, this complex is the same as Farley’s.

The Cubical Complex

Theorem (B-F)
Given an expanding replacement rule e -> R and base graph (Subscript[G, 0]): 
  1.  The resulting complex K is a CAT(0) cubical complex.
  2.  The rearrangement group acts on K by isometries.
  3.  Vertex stabilizers for this action are finite.

Morse Function φ:K->[0,∞)
    •  For a vertex X((Subscript[G, 0]))->X(G), let φ(vertex)= # of edges of G.
    •  Extend linearly to cubes.

Some Technology

There is a “standard” procedure for proving that a group has type (Subscript[F, ∞]) (due to Brown, Bestvina, Brady, etc.):

✓ 1.  Find a nice action of the group on a complex K. 
✓ 2.  Choose an appropriate Morse function φ: K -> [0, ∞). 
 -> 3.  Analyze the connectivity of the descending links.

Descending Links

Descending Links

Each vertex X((Subscript[G, 0]))->X(G) in K has a descending link.
    •  Always a simplicial flag complex
    •  Depends only on the isomorphism type of G.

What the Technology Tells Us
If the descending link is n-connected for all but finitely many graphs G, then the rearrangement group has type (Subscript[F, n+1]).

If this holds for all n, then the group has type (Subscript[F, ∞]).

Descending Links

For a given graph G:

There is one vertex in the descending link for each merge (inverse split) starting at G.

Descending Links

For a given graph G:

There is one vertex in the descending link for each merge (inverse split) starting at G.

Descending Links

For a given graph G:

There is one vertex in the descending link for each merge (inverse split) starting at G.

Descending Links

For a given graph G:

There is one vertex in the descending link for each merge (inverse split) starting at G.

Descending Links

For a given graph G:

Two vertices are connected by an edge if the corresponding merges are disjoint.

Descending Links

For a given graph G:

Two vertices are connected by an edge if the corresponding merges are disjoint.

Descending Links

Here’s a possible graph for Thompson’s group F:

The descending link has five vertices:

Descending Links

Here’s a possible graph for Thompson’s group F:

The descending link has five vertices:

Descending Links

Here’s a possible graph for Thompson’s group F:

The descending link has five vertices:

A Useful Criterion

Definition
A flag complex Δ is k-dense if every vertex of Δ is connected to all but at most k other vertices.

For example, the descending links for F are 2-dense.

Theorem (B-F)
A k-dense flag complex with at least k(k+1)(n+1)+1 vertices
is n-connected.

Example. Thompson’s group F has type (Subscript[F, ∞]).

The Vicsek Fractal

The Vicsek Fractal

Descending links for the Vicsek fractal are 3-dense.

So it suffices to show that almost all descending links have a large number of vertices, i.e. large graphs G have a large number of merges.

This isn’t true for the Basilica:

The Vicsek Fractal

For the Vicsek fractal, the graphs G look like trees of “plusses”:

The Vicsek Fractal

A tree with n plusses has at least n / 2 possible merges:

The Vicsek Fractal

Thus the Vicsek rearrangement group has type (Subscript[F, ∞]).

The End