Self-Similar
Groups
A Penland
Self-Similar Groups and Groups Generated
by Automata
Andrew Penland
apenland@math.tamu.edu
September 2, 2014
Basic Definitions - Words and Trees
Self-Similar
Groups
A Penland
X - finite set (e.g. {0, 1})
X n = words of length n in X
X [d] =
d
[
X i = words of length d or less
i=0
X (d)
=
d−1
[
i=0
X i = words of length less than d
Basic Definitions - Words and Trees
Self-Similar
Groups
A Penland
X - finite set (e.g. {0, 1})
X n = words of length n in X
X [d] =
d
[
X i = words of length d or less
i=0
X (d)
=
d−1
[
i=0
X i = words of length less than d
Basic Definitions - Words and Trees
Self-Similar
Groups
A Penland
X - finite set (e.g. {0, 1})
X n = words of length n in X
X [d] =
d
[
X i = words of length d or less
i=0
X (d)
=
d−1
[
i=0
X i = words of length less than d
Basic Definitions - Words and Trees
Self-Similar
Groups
A Penland
X - finite set (e.g. {0, 1})
X n = words of length n in X
X [d] =
d
[
X i = words of length d or less
i=0
X (d)
=
d−1
[
i=0
X i = words of length less than d
Basic Definitions - Words and Trees
Self-Similar
Groups
A Penland
X - finite set (e.g. {0, 1})
X n = words of length n in X
X [d] =
d
[
X i = words of length d or less
i=0
X (d)
=
d−1
[
i=0
X i = words of length less than d
Words and Trees
Self-Similar
Groups
A Penland
X (d) ⇐⇒ |X |-ary tree with d levels
X [d] ⇐⇒ |X |-ary tree with d + 1 levels
0
00
000
1
01
001 010
10
011 100
Figure: X [3] = X (4)
11
101 110
111
Self-Similar
Groups
A Penland
X∗ =
∞
[
X i = all words in X
i=0
∼
= infinite |X |-regular tree rooted at ..
.
Vertices - elements of X ∗
Edges - {w, wx}(w ∈ X ∗ , x ∈ X )
Self-Similar
Groups
A Penland
X∗ =
∞
[
X i = all words in X
i=0
∼
= infinite |X |-regular tree rooted at ..
.
Vertices - elements of X ∗
Edges - {w, wx}(w ∈ X ∗ , x ∈ X )
Self-Similar
Groups
A Penland
X∗ =
∞
[
X i = all words in X
i=0
∼
= infinite |X |-regular tree rooted at ..
.
Vertices - elements of X ∗
Edges - {w, wx}(w ∈ X ∗ , x ∈ X )
Automorphisms of X ∗
Self-Similar
Groups
A Penland
Definition
An automorphism of X ∗ is a graph automorphism of the
rooted tree X ∗ .
Aut(X ∗ ) - group of all automorphisms of X ∗
Groups generated by automata – special subgroups of
Aut(X ∗ )
Automorphisms of X ∗
Self-Similar
Groups
A Penland
Definition
An automorphism of X ∗ is a graph automorphism of the
rooted tree X ∗ .
Aut(X ∗ ) - group of all automorphisms of X ∗
Groups generated by automata – special subgroups of
Aut(X ∗ )
Automorphisms of X ∗
Self-Similar
Groups
A Penland
Definition
An automorphism of X ∗ is a graph automorphism of the
rooted tree X ∗ .
Aut(X ∗ ) - group of all automorphisms of X ∗
Groups generated by automata – special subgroups of
Aut(X ∗ )
Automorphisms of X ∗
Self-Similar
Groups
A Penland
Definition
An automorphism of X ∗ is a graph automorphism of the
rooted tree X ∗ .
Aut(X ∗ ) - group of all automorphisms of X ∗
Groups generated by automata – special subgroups of
Aut(X ∗ )
Automata
Self-Similar
Groups
A Penland
Definition
A non-initial Mealy automaton A is a 4-tuple
A = (S, X , P, T ) where
S - set of states
X - alphabet
P : S × X → X - print function
T : S × X → S - transition function
Automata
Self-Similar
Groups
A Penland
s1
x |y
Indicates P(s1 , x) = y and T (s1 , x) = s2 .
s2
Self-Similar
Groups
A Penland
Automaton - used as “machine” which transforms elements
of X ∗
For each s ∈ S, there is an induced function Ts : X → X
given by Ts (x) = T (s, x)
Choosing an initial state s0 ∈ S, Ts0 extends to a function
X ∗ → X ∗.
Self-Similar
Groups
A Penland
Automaton - used as “machine” which transforms elements
of X ∗
For each s ∈ S, there is an induced function Ts : X → X
given by Ts (x) = T (s, x)
Choosing an initial state s0 ∈ S, Ts0 extends to a function
X ∗ → X ∗.
Self-Similar
Groups
A Penland
Automaton - used as “machine” which transforms elements
of X ∗
For each s ∈ S, there is an induced function Ts : X → X
given by Ts (x) = T (s, x)
Choosing an initial state s0 ∈ S, Ts0 extends to a function
X ∗ → X ∗.
States Give Functions – Example
Self-Similar
Groups
A Penland
1|0
a
0|1
e
States Give Functions – Example
Self-Similar
Groups
A Penland
1|0
0|1
a
e
Ta (00) = 10
Te (00) =
Ta (10) = 01
Te (10) =
Ta (01) = 11
Te (01) =
Ta (11) = 00
Te (11) =
States Give Functions – Example
Self-Similar
Groups
A Penland
1|0
0|1
a
e
Ta (00) = 10
Te (00) =
Ta (10) = 01
Te (10) =
Ta (01) = 11
Te (01) =
Ta (11) = 00
Te (11) =
States Give Functions – Example
Self-Similar
Groups
A Penland
1|0
0|1
a
e
Ta (00) = 10
Te (00) =
Ta (10) = 01
Te (10) =
Ta (01) = 11
Te (01) =
Ta (11) = 00
Te (11) =
States Give Functions – Example
Self-Similar
Groups
A Penland
1|0
0|1
a
e
Ta (00) = 10
Te (00) =
Ta (10) = 01
Te (10) =
Ta (01) = 11
Te (01) =
Ta (11) = 00
Te (11) =
States Give Functions – Example
Self-Similar
Groups
A Penland
1|0
0|1
a
e
Ta (00) = 10
Te (00) =
Ta (10) = 01
Te (10) =
Ta (01) = 11
Te (01) =
Ta (11) = 00
Te (11) =
States Give Functions – Example
Self-Similar
Groups
A Penland
1|0
0|1
a
e
Ta (00) = 10
Te (00) =
Ta (10) = 01
Te (10) =
Ta (01) = 11
Te (01) =
Ta (11) = 00
Te (11) =
States Give Functions – Example
Self-Similar
Groups
A Penland
1|0
0|1
a
e
Ta (00) = 10
Te (00) =
Ta (10) = 01
Te (10) =
Ta (01) = 11
Te (01) =
Ta (11) = 00
Te (11) =
States Give Functions – Example
Self-Similar
Groups
A Penland
1|0
0|1
a
e
Ta (00) = 10
Te (00) =
Ta (10) = 01
Te (10) =
Ta (01) = 11
Te (01) =
Ta (11) = 00
Te (11) =
States Give Functions – Example
Self-Similar
Groups
A Penland
1|0
0|1
a
e
Ta (00) = 10
Te (00) =
Ta (10) = 01
Te (10) =
Ta (01) = 11
Te (01) =
Ta (11) = 00
Te (11) =
{0, 1}2 ⇐⇒ Z/4Z
Self-Similar
Groups
A Penland
00 ⇐⇒ 0
10 ⇐⇒ 1
01 ⇐⇒ 2
11 ⇐⇒ 3
Ta : Z/4Z → Z/4Z
Self-Similar
Groups
A Penland
Ta (00) = 10
Ta (10) = 01
Ta (01) = 11
Ta (11) = 00
Ta : Z/4Z → Z/4Z
Self-Similar
Groups
A Penland
Ta (00) = 10
Ta (10) = 01
Ta (01) = 11
Ta (11) = 00
Ta : Z/4Z → Z/4Z
Self-Similar
Groups
A Penland
Ta (00) = 10
Ta (10) = 01
Ta (01) = 11
Ta (11) = 00
Ta : Z/4Z → Z/4Z
Self-Similar
Groups
A Penland
Ta (00) = 10
Ta (10) = 01
Ta (01) = 11
Ta (11) = 00
Ta : Z/4Z → Z/4Z
Self-Similar
Groups
A Penland
Ta (00) = 10
Ta (10) = 01
Ta (01) = 11
Ta (11) = 00
{0, 1}2 ⇐⇒ Z/4Z
Self-Similar
Groups
A Penland
Ta (0) = 1
Ta (1) = 2
Ta (2) = 3
Ta (3) = 0
Under this identification, Ta (n) = Ta (n + 1) mod 4.
{0, 1}n ⇐⇒ Z/2n Z
Self-Similar
Groups
A Penland
For larger powers of 2.
{0, 1}n ⇐⇒ Z/2n Z
x0 x1 x2 . . . xn−1 ⇐⇒
n−1
X
!
xi 2
i
mod n.
i=0
Under this identification Ta (x) = (x + 1) mod 2n .
{0, 1}n ⇐⇒ Z/2n Z
Self-Similar
Groups
A Penland
For larger powers of 2.
{0, 1}n ⇐⇒ Z/2n Z
x0 x1 x2 . . . xn−1 ⇐⇒
n−1
X
!
xi 2
i
mod n.
i=0
Under this identification Ta (x) = (x + 1) mod 2n .
Odometer
Self-Similar
Groups
A Penland
The group generated by {Ta , Te } is a subgroup of Aut(X ∗ )
called the odometer group.
Isomorphic to Z.
Odometer
Self-Similar
Groups
A Penland
The group generated by {Ta , Te } is a subgroup of Aut(X ∗ )
called the odometer group.
Isomorphic to Z.
Odometer
Self-Similar
Groups
A Penland
The group generated by {Ta , Te } is a subgroup of Aut(X ∗ )
called the odometer group.
Isomorphic to Z.
Groups Generated by Automata
Self-Similar
Groups
A Penland
Proposition
Let A = (S, X , P, T ) be a non-initial Mealy automaton. If for
each s ∈ S, the function Ts : X → X is a permutation of X ,
then the functions {Ts }s∈S generate a subgroup of Aut(X ∗ ).
We usually refer to this subgroup as the group generated by
A.
We also usually refer to s instead of Ts .
Groups Generated by Automata
Self-Similar
Groups
A Penland
Proposition
Let A = (S, X , P, T ) be a non-initial Mealy automaton. If for
each s ∈ S, the function Ts : X → X is a permutation of X ,
then the functions {Ts }s∈S generate a subgroup of Aut(X ∗ ).
We usually refer to this subgroup as the group generated by
A.
We also usually refer to s instead of Ts .
Groups Generated by Automata
Self-Similar
Groups
A Penland
Proposition
Let A = (S, X , P, T ) be a non-initial Mealy automaton. If for
each s ∈ S, the function Ts : X → X is a permutation of X ,
then the functions {Ts }s∈S generate a subgroup of Aut(X ∗ ).
We usually refer to this subgroup as the group generated by
A.
We also usually refer to s instead of Ts .
Groups Generated by Automata
Self-Similar
Groups
A Penland
Proposition
Let A = (S, X , P, T ) be a non-initial Mealy automaton. If for
each s ∈ S, the function Ts : X → X is a permutation of X ,
then the functions {Ts }s∈S generate a subgroup of Aut(X ∗ ).
We usually refer to this subgroup as the group generated by
A.
We also usually refer to s instead of Ts .
Groups Generated by Automata
Self-Similar
Groups
A Penland
Proposition
Let A = (S, X , P, T ) be a non-initial Mealy automaton. If for
each s ∈ S, the function Ts : X → X is a permutation of X ,
then the functions {Ts }s∈S generate a subgroup of Aut(X ∗ ).
We usually refer to this subgroup as the group generated by
A.
We also usually refer to s instead of Ts .
Groups Generated by Automata
Self-Similar
Groups
A Penland
Proposition
Let A = (S, X , P, T ) be a non-initial Mealy automaton. If for
each s ∈ S, the function Ts : X → X is a permutation of X ,
then the functions {Ts }s∈S generate a subgroup of Aut(X ∗ ).
We usually refer to this subgroup as the group generated by
A.
We also usually refer to s instead of Ts .
The Grigorchuk Group
Self-Similar
Groups
A Penland
The famed Grigorchuk group is another example of a group
generated by an automaton.
“There are other mathematical objects which play a really
important role in their subject: the hyperbolic plane for
spaces of negative curvature, SL(2, R) for representations
of semi-simple groups, or the hyperfinite II1 factor for von
Neumann algebras...In our opinion, the Grigorchuk group is
a good candidate for membership in this club.”
-Pierre de la Harpe,
Topics In Geometric Group Theory
The Grigorchuk Group
Self-Similar
Groups
A Penland
The famed Grigorchuk group is another example of a group
generated by an automaton.
“There are other mathematical objects which play a really
important role in their subject: the hyperbolic plane for
spaces of negative curvature, SL(2, R) for representations
of semi-simple groups, or the hyperfinite II1 factor for von
Neumann algebras...In our opinion, the Grigorchuk group is
a good candidate for membership in this club.”
-Pierre de la Harpe,
Topics In Geometric Group Theory
Problems Solved By Grigorchuk Group
Self-Similar
Groups
A Penland
Burnside Problem - provide examples of infinite groups
which are finitely generated such that each element
has finite order.
First example of group of intermediate growth
(answered 1968 question of Milnor)
First example of amenable but not elementary
amenable group. (answered 1957 question of Day)
Led to introduction of self-similarity in group theory.
Problems Solved By Grigorchuk Group
Self-Similar
Groups
A Penland
Burnside Problem - provide examples of infinite groups
which are finitely generated such that each element
has finite order.
First example of group of intermediate growth
(answered 1968 question of Milnor)
First example of amenable but not elementary
amenable group. (answered 1957 question of Day)
Led to introduction of self-similarity in group theory.
Problems Solved By Grigorchuk Group
Self-Similar
Groups
A Penland
Burnside Problem - provide examples of infinite groups
which are finitely generated such that each element
has finite order.
First example of group of intermediate growth
(answered 1968 question of Milnor)
First example of amenable but not elementary
amenable group. (answered 1957 question of Day)
Led to introduction of self-similarity in group theory.
Problems Solved By Grigorchuk Group
Self-Similar
Groups
A Penland
Burnside Problem - provide examples of infinite groups
which are finitely generated such that each element
has finite order.
First example of group of intermediate growth
(answered 1968 question of Milnor)
First example of amenable but not elementary
amenable group. (answered 1957 question of Day)
Led to introduction of self-similarity in group theory.
Problems Solved By Grigorchuk Group
Self-Similar
Groups
A Penland
Burnside Problem - provide examples of infinite groups
which are finitely generated such that each element
has finite order.
First example of group of intermediate growth
(answered 1968 question of Milnor)
First example of amenable but not elementary
amenable group. (answered 1957 question of Day)
Led to introduction of self-similarity in group theory.
Self-Similarity
Self-Similar
Groups
A Penland
wX ∗ ∼
= X ∗ for any word w.
Sections
Self-Similar
Groups
A Penland
For each g ∈ Aut(X ∗ ) and w ∈ X ∗ , there exists a unique
element gw such that g(wv ) = g(w)gw (v ) for all v ∈ X ∗ .
This element gv is called the section of g at v .
Self-Similar Groups
Self-Similar
Groups
A Penland
Definition
A self-similar group is a subgroup of Aut(X ∗ ) which contains
all sections of all of its elements.
Various properties of self-similar groups have received
attention in the last 20 years.
Self-Similar Groups
Self-Similar
Groups
A Penland
Definition
A self-similar group is a subgroup of Aut(X ∗ ) which contains
all sections of all of its elements.
Various properties of self-similar groups have received
attention in the last 20 years.
Big Idea of Rest of Talk
Self-Similar
Groups
A Penland
The remainder of this talk is about the Hausdorff dimension
of finitely constrained groups of binary tree automorphisms.
Henceforth, X = {0, 1}. Thus
Sym(X ) ∼
= C2 = {id, σ} = {◦, •}.
G = Aut(X ∗ ).
Activity
Self-Similar
Groups
A Penland
Given g ∈ G, the root activity α : G → C2 is defined by
(
◦, g(0) = 0
α(g) =
.
•, g(0) = 1
For w ∈ X ∗ , the activity at w is α(gw ).
Portraits
Self-Similar
Groups
A Penland
∗
G ⇐⇒ (C2 )(X ) via the portrait map [τ (g)](w) = α(gw )
Abusing notation, we write [τ (g)](w) as g(w) .
..
.
..
.
The Group G(d)
Self-Similar
Groups
A Penland
We define the group G(d) = Aut(X [d] ).
Note that there is a natural homomorphism πd : G → G(d).
G(d) ∼
= Sylow 2 − subgroup of Sym(2d )
∼
= C2 o C2 o . . . o C2
|
{z
}
d
Profinite Group
Self-Similar
Groups
A Penland
G is a profinite (pro-2) topological group as the inverse limit
of the inverse system
. . . → G(3) → G(2) → G(1) ∼
= C2 .
G as a metric space
Self-Similar
Groups
A Penland
For g, h ∈ G, we have
(
0, g = h
d(g, h) =
1
, g 6= h
22n −1
where
n = inf{k | there exists u ∈ X k s.t.g(u) 6= h(u) }
= inf{k | πk (g) 6= πk (h)}).
Algebra of Continuous Functions on Aut(X ∗ )
Self-Similar
Groups
A Penland
A - algebra of continuous functions on Aut(X ∗ ) (pointwise
addition and multiplication)
Given w ∈ X ∗ , we define a function [w] : Aut(X ∗ ) → C2 by
[w](g) = α(gw ).
Proposition (Siegenthaler, 2009)
Every element of A is a polynomial in a finite number of
variables [v ], v ∈ X ∗ .
Zariski Topology
Self-Similar
Groups
A Penland
Proposition (Siegenthaler, 2009)
The topological structure of Aut(X ∗ ) is equivalent to a
Zariski topology coming from A.
Hausdorff Dimension
Self-Similar
Groups
A Penland
Hausdorff dimension - a topological dimension defined for
any metric space.
With the metric introduced above, we can naturally
associate a Hausdorff dimension to closed subgroups of G.
Hausdorff Dimension of Pro-p Groups
Self-Similar
Groups
A Penland
Abercrombie, 1994
Barnea and Shalev, 1997
Abert and Virag, 2005 – for any λ ∈ [0, 1], there exists a
subgroup of Aut(X ∗ ) with Hausdorff dimension λ
Portraits of G(d)
Self-Similar
Groups
A Penland
G(d) ⇐⇒ (C2 )X
(d)
established by portrait map.
(
◦, if g(w0) = g(w)0
[α(g)](w) =
•, if g(w0) = g(w)1
Tree Automorphisms and Patterns
Self-Similar
Groups
A Penland
Definition
A pattern of size d is an element of G(d). A pattern group of
pattern size d is a subgroup of G(d).
Tree Automorphisms and Patterns
Self-Similar
Groups
A Penland
Definition
Given a pattern p ∈ G(d) and g ∈ G, we say p appears in g
if there exists w ∈ X ∗ such that g(wv ) = p(v ) for all v ∈ X (d) .
Finitely Constrained Groups
Self-Similar
Groups
A Penland
Definition
Let P be a proper nontrivial pattern group with pattern size
d. We define
GP = {g ∈ Aut(X ∗ ) | p appears in g =⇒ p ∈ P}.
We say GP is defined by P.
Definition
Let H ≤ G = Aut(X ∗ ). We say H is a finitely constrained
group if there exists P G(d) such that H is defined by P.
Any finitely constrained group is a closed, self-similar
subgroup of G.
Finitely Constrained Groups
Self-Similar
Groups
A Penland
Definition
Let P be a proper nontrivial pattern group with pattern size
d. We define
GP = {g ∈ Aut(X ∗ ) | p appears in g =⇒ p ∈ P}.
We say GP is defined by P.
Definition
Let H ≤ G = Aut(X ∗ ). We say H is a finitely constrained
group if there exists P G(d) such that H is defined by P.
Any finitely constrained group is a closed, self-similar
subgroup of G.
Finitely Constrained Groups
Self-Similar
Groups
A Penland
Definition
Let P be a proper nontrivial pattern group with pattern size
d. We define
GP = {g ∈ Aut(X ∗ ) | p appears in g =⇒ p ∈ P}.
We say GP is defined by P.
Definition
Let H ≤ G = Aut(X ∗ ). We say H is a finitely constrained
group if there exists P G(d) such that H is defined by P.
Any finitely constrained group is a closed, self-similar
subgroup of G.
Hausdorff Dimension of Finitely Constrained
Groups
Self-Similar
Groups
A Penland
If P ≤ G(d),
2
1
, 2d−1
,...,1 −
dimH (GP ) ∈ {0, 2d−1
(Bartholdi, 2006; Šunić, 2007)
2
,1
2d−1
−
1
, 1}
2d−1
Lemma (P, 2014)
Given any d ∈ N and any 1 ≤ k ≤ 2d−1 , there exists a
k
finitely constrained group with Hausdorff dimension 1 − 2d−1
.
Hausdorff Dimension of Finitely Constrained
Groups
Self-Similar
Groups
A Penland
If P ≤ G(d),
2
1
, 2d−1
,...,1 −
dimH (GP ) ∈ {0, 2d−1
(Bartholdi, 2006; Šunić, 2007)
2
,1
2d−1
−
1
, 1}
2d−1
Lemma (P, 2014)
Given any d ∈ N and any 1 ≤ k ≤ 2d−1 , there exists a
k
finitely constrained group with Hausdorff dimension 1 − 2d−1
.
Topological Finite Generation
Self-Similar
Groups
A Penland
Definition
A subgroup of G is called topologically finitely generated if it
is the topological closure of a finitely generated group.
Question
For pattern size d and 1 ≤ k ≤ 2d−1 , does there exist a
topologically finitely generated, finitely constrained group of
k
?
Hausdorff dimension 1 − 2d−1
January 2014
Self-Similar
Groups
A Penland
1−
1
2d−1
1−
2
2d−1
1−
3
2d−1
2
Šunić, 2011
N/A
N/A
3
B-S, 2013
B-S, 2013
B-S, 2013
4
B-S, 2013
B-S, 2013
Grigorchuk,2000
5
?
?
···
Šunić, 2007
d >5
?
?
Šunić, 2007
Note: B - S = Bondarenko and Samoilovych
May 2014
Self-Similar
Groups
A Penland
1−
1
2d−1
1−
2
2d−1
1−
3
2d−1
2
Šunić, 2011
N/A
N/A
3
B-S, 2013
B-S, 2013
B-S, 2013
4
B-S, 2013
B-S, 2013
Grigorchuk,2000
5
P-Šunić, 2014
?
Šunić, 2007
P-Šunić, 2014
?
Šunić, 2007
d >5
···
Now
Self-Similar
Groups
A Penland
1−
1
2d−1
1−
2
2d−1
1−
3
2d−1
2
Šunić, 2011
N/A
N/A
3
B-S, 2013
B-S, 2013
B-S, 2013
4
B-S, 2013
B-S, 2013
Grigorchuk,2000
5
P-Šunić, 2014
P, 2014
Šunić, 2007
d >5
P-Šunić, 2014
?
Šunić, 2007
···