Self-similar group actions and KMS states
Michael Whittaker
(University of Wollongong)
Joint with Marcelo Laca, Iain Raeburn, and Jacqui Ramagge
Pure Mathematics Seminar
(University of New South Wales)
12 November 2013
Self-similar actions
Suppose X is a finite set of cardinality |X |;
let X n denote
[ the set of words of length n in X ,
∗
X n.
let X =
n∈N
Definition
A faithful action of a group G on X ∗ is a self-similar group action
if, for all g ∈ G and x ∈ X , there exist unique g |x ∈ G such that
g · (xw ) = (g · x)(g |x · w )
for all finite words w ∈ X ∗ .
The group element g |x is called the restriction of g to x.
We may replace the letter x by an initial word v : for g ∈ G and
v ∈ X k there exists a unique g |v ∈ G such that
g · (vw ) = (g · v )(g |v · w )
with
and
for all w ∈ X ∗ .
g · v = (g · v1 )(g |v1 · v2 ) · · · (g |v1 |v2 ··· |vk−1 · vk )
g |v = (g |v1 )|v2 · · · |vk
Restrictions
Lemma
Suppose (G , X ) is a self-similar action. Restrictions satisfy
g |vw = (g |v )|w ,
gh|v = g |h·v h|v ,
−1
g |−1
|g ·v
v =g
for all g , h ∈ G and v , w ∈ X ∗ .
Proof of gh|v = g |h·v h|v .
For all w ∈ X ∗ we have
gh · (vw ) = (gh · v )(gh|v · w )
gh · (vw ) = g · (h · v )(h|v · w ) = (gh · v )(g |h·v h|v · w ).
Since the action of G on X ∗ is faithful, we have
gh|v = g |h·v h|v .
Example: the odometer action
Let X = {0, 1} and G = Z
Let g denote the generator 1 ∈ Z
(Z, X ) is a self-similar action described by:
g · 0w = 1w
g · 1w = 0(g · w )
for every finite word w ∈ X ∗
For example, g 3 denotes 3 ∈ Z and acts on the word 01100 by
g 3 · 01100 = g 2 · 11100 = g · 00010 = 10010.
The odometer continued
X∗
g · X∗
∅
∅
0
00
1
01
10
1
11
000 001 010 011 100 101 110 111
.. .. .. .. .. .. .. ..
. . . . . . . .
10
0
11
01
00
100 101 110 111 010 011 001 000
.. .. .. .. .. .. .. ..
. . . . . . . .
Figure: The odometer action
Contracting SSAs, nucleus and Moore diagrams
(G , X ) is contracting if there is a finite S ⊂ G such that for
every g ∈ G there exists n ∈ N with
{g |v : v ∈ X ∗ , |v | ≥ n} ⊂ S.
The nucleus of a contracting (G , X ) is the smallest such S:
N :=
∞
[ \
{g |v : v ∈ X ∗ , |v | ≥ n}.
g ∈G n=0
For g ∈ S (S ⊂ G closed under restriction), the Moore
diagram with vertex set S has a directed edge
(x, y )
g −−−−→ h = g |x for each self similarity relation
g · (xw ) = y (h · w ).
The Moore diagram for the nucleus of the odometer
The nucleus of the odometer action is N = {e, g , g −1 }.
(1,1)
(1,0)
(0,1)
−g
(0,1)
e
(0,0)
g
(1,0)
The Grigorchuk group (1980)
Let X = {x, y }
Consider the rooted homogeneous tree TX with vertex set X ∗ .
Grigorchuk group is generated by four automorphisms
a, b, c, d of TX defined recursively by
a · xw = yw
b · xw = x(a · w )
c · xw = x(a · w )
d · xw = xw
a · yw = xw
b · yw = y (c · w )
c · yw = y (d · w )
d · yw = y (b · w ).
Proposition
The generators a, b, c, d of G all have order two, and satisfy
cd = b = dc, db = c = bd and bc = d = cb. The self-similar
action (G , X ) is contracting with nucleus N = {e, a, b, c, d}.
The Moore diagram for the nucleus of the Grigorchuk
group
(x,x)
a
c
(y ,y )
(x,x)
(x,y )
(y ,x)
(y ,y )
b
(y ,y )
(y ,y )
e
d
(x,x)
(x,x)
Properties of the Grigorchuk group
Theorem (Grigorchuk 1980)
The Grigorchuk group is a finitely generated infinite 2-torsion
group.
(This gives a particularly nice example of a Burnside group)
Theorem (Grigorchuk 1984)
The Grigorchuk group has intermediate growth.
(Solved a Milnor problem from 1968)
The basilica group [Grigorchuk and Żuk 2003]
Let X = {x, y }
Consider the rooted homogeneous tree TX with vertex set X ∗ .
Two automorphisms a and b of TX are recursively defined by
a · (xw ) = y (b · w )
a · (yw ) = xw
b · (xw ) = x(a · w )
b · (yw ) = yw
for w ∈ X ∗ .
The basilica group B is the subgroup of Aut TX generated by
{a, b}. The pair (B, X ) is then a self-similar action.
The nucleus is N = {e, a, b, a−1 , b −1 , ba−1 , ab −1 }.
The Moore diagram for the nucleus of the basilica group
b −1
b
(y ,y )
(x,y )
(y ,y )
(x,x)
(y ,x)
(x,x)
a−1
a
(y ,x)
(x,y )
e
(x,x)
(y ,x)
(y ,y )
(y ,x)
ab −1
(x,y )
ba−1
(x,y )
Properties of the basilica group
Theorem (Grigorchuk and Żuk 2003)
The basilica group
is torsion free,
has exponential growth,
has no free non-abelian subgroups,
is not elementary amenable.
Theorem (Bartholdi and Virág 2005)
The basilica group is amenable.
C ∗ -algebras
A C ∗ -algebra is a Banach ∗-algebra A such that for all a in A,
kaa∗ k = kak2
Examples: C, Mn (C), C0 (X ), B(H), K(H), ...
Any element U ∈ A such that U ∗ U = UU ∗ = 1 is called a
unitary and any element S ∈ A such that S ∗ S = 1 is called an
isometry.
The C*-algebras T (G , X ) and O(G , X )
Let (G , X ) be a self-similar action.
T (G , X ) := universal C ∗ -algebra with generators {Sx : x ∈ X }
and {Ug : g ∈ G } such that
(
1 if x =y
∗
(T1) Sy Sx =
S
T|X |
0 if x 6= y
(T2) Ug Uh = Ugh ;
Ug∗ = Ug −1 ;
(T3) Ug Sx = Sg ·x Ug |x
Ue = 1
U
C ∗ (G )
self-similarity comm. rels.
g · (xw ) = (g · x)(g |x · w )
O(G , X ) := the universal Toeplitz algebra T (G , X ) with the extra
relation
P
∗
(O)
S
O|X |
x∈X Sx Sx = 1
Remark: Nekrashevych gave the first definition of O(G , X ).
Spanning set and dynamics
For a word v = x1 x2 · · · xn , we let Sv := Sx1 Sx2 · · · Sxn .
T (G , X ) = span{Sv Ug Sw∗ : v , w ∈ X ∗ , g ∈ G }.
If (G , X ) is contracting,
O(G , X ) = span{Sv Ug Sw∗ : v , w ∈ X ∗ , g ∈ N }
The dynamics on T (G , X ), and on O(G , X ) are defined by
σt (Sv Ug Sw∗ ) = e t(|v |−|w |) Sv Ug Sw∗
Interested in (KMS) equilibrium states of (T (G , X ), σ) and of
(O(G , X ), σ).
KMS states
Given a continuous action σ : R → Aut(A), there is a dense
*-subalgebra of σ-analytic elements: t 7→ σt (a) extends to an
entire function z 7→ σz (a).
Definition
The state ϕ of A satisfies the KMS condition at inverse
temperature β ∈ [0, ∞) if
ϕ(ab) = ϕ(b σiβ (a))
whenever a and b are analytic for σ.
Note: it suffices to verify the above for elements that span a
dense subalgebra, e.g, in our case, the spanning set {Sv Ug Sw∗ }
Properties of KMS states:
Haag-Hugenholtz-Winnink proposed the KMS condition as a
definition of equilibrium for quantum systems.
KMS states are a noncommutative phenomenon, If A has a
faithful KMS state and A is commutative, then σ is trivial.
If β 6= 0 and ϕ is a KMSβ -state, then ϕ is σ-invariant.
KMS states have a natural notion of a phase transition (an
abrupt change in the physical properties of a system).
Theorem (Laca-Raeburn-Ramagge-W ’13)
1. If β ∈ [0, log |X |), there are no KMSβ states for σ;
2. if β ∈ (log |X |, ∞], for each normalized trace τ on C ∗ (G )
define ψβ,τ (Sv Ug Sw∗ ) = 0 if v 6= w , and
ψβ,τ (Sv Ug Sv∗ ) = (1 − |X |e −β )
∞
X
e −β(k+|v |)
k=0
X
τ (δg |y )
y ∈X k
g ·y =y
the map τ 7→ ψβ,τ is a homeomorphism from the normalised
traces on C ∗ (G ) to the KMSβ states of T (G , X ).
3. the KMSlog |X | states of T (G , X ) arise from KMS states of
O(G , X ); and there is at least this one:
(
|X |−|v | cg if v = w
∗
ψlog |X | (Sv Ug Sw ) =
0
otherwise.
If (G , X ) is contracting, this is the only one.
The asymptotic proportion of points fixed by g ∈ G
Let τ be the usual trace on C ∗ (G ), i.e. τ (δg ) = 0 unless g = e;
then let β & log |X |,
ψβ,τ (ug ) = (1 − |X |e
−β
)
∞
X
e −βk
k=0
X
τ (δg |y ) −→ ??
y ∈X k
g ·y =y
For each k ∈ N and g ∈ G define
Fgk := {y ∈ X n : g · y = y and g |y = e}.
Then
{|Fgk ||X |−k } % cg ∈ [0, 1) and since
X
τ (δg |y ) = |Fgk |,
y ∈X k
g ·y =y
the above limit is also cg .
How do we actually compute cg ?
Calculating cg using the Moore diagram
To calculate values of the KMS states explicitly, we need to
evaluate the limit
cg = lim |Fgk ||X |−k
k→∞
Each v ∈ Fgk corresponds to a path µv in the Moore diagram:
µv :=g
(v1 ,v1 )
g |v1
(v2 ,v2 )
g |v1 v2
(v3 ,v3 )
···
Notice that all the labels have the form (x, x).
Every path with labels (x, x) arises this way.
(vk ,vk )
g |v = e
Example: the odometer action
Proposition
The C ∗ -algebra O(Z, X ) has a unique KMSlog 2 state, which is
given on the nucleus N = {e, g , g −1 } by
(
1 for n = e
φ(Un ) =
0 for n = g , g −1
Sketch.
(1,1)
(1,0)
(0,1)
−g
(1,1)
(0,1)
e
(0,0)
g
(1,0)
reduces to
e
(0,0)
The Grigorchuk group
Proposition
Let (G , X ) be the self-similar action of the Grigorchuk group.
Then (O(G , X ), σ) has a unique KMSlog 2 state ψ which is given
on the nucleus N = {e, a, b, c, d} by


1
for g = e




0
for g = a

ψ(Ug ) = 1/7 for g = b



2/7 for g = c




4/7 for g = d.
The Grigorchuk group
Sketch of proof.
(x,x)
a
c
c
(x,x)(y ,y )
(x,y )
(y ,y )
(y ,x) b
(y ,y )
reduces to
(y ,y )
(y ,y )
e
(y ,y )
d
(y ,y )
e
(x,x)
(x,x)
(y ,y )
b
d
(x,x)
(x,x)
Computation of cd for the Grigorchuk group
d
e
b
e
e
e
e
e
e
e
a
e
e
e
e
e
e
c
a
e
2−1
d
e
b
..
.
2−4
2−7
∞
1 X 1 3n
1 1 4
cd =
=
= .
2
2
2 1 − 18
7
n=0
The basilica group
Proposition
The C ∗ -algebra O(B, X ) has a unique KMSlog 2 state, which is
given on the nucleus N = {e, a, b, a−1 , b −1 , ab −1 , ba−1 } by


1 for g = e
ψ(ug ) = 21 for g = b, b −1


0 for g = a, a−1 , ab −1 , ba−1 .
The basilica group
Sketch of proof.
(x,y )
a
b
(y ,y )
(y ,y )
(x,x)
b −1
(x,x)
(y ,x)
(x,x)
e
(y ,x)
(y ,x)
ab −1
(x,y )
(y ,y )
(x,y )
ba−1
(x,y )
b
(y ,x)
a−1
(y ,y )
reduces to
(y ,y )
b −1
e
(x,x)
(y ,y )
Computation of cb for the basilica group
b
a
e
e
e
e
e
e
e
e
e
e
e
2−1
cb =
1
2
e
e
e
e
References:
1. M. Laca, I. Raeburn, J. Ramagge, and M. Whittaker
Equilibrium states on the Cuntz-Pimsner algebras
of self-similar actions, ArXiv:1301.4722 (2013).
2. V. Nekrashevych, C ∗ -algebras and self-similar groups, J. reine
angew. Math. 630 (2009), 59–123.
3. V. Nekrashevych, Self-similar groups, Mathematical Surveys
and Monographs vol. 117,. Amer. Math. Soc., Providence,
RI, 2005.