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Spectral properties related to spinal groups
Aitor Perez
University of Geneva
February 2019
Groups, automata and graphs, TUGraz
Outline
Spinal groups Schreier graphs Spectral properties
Grigorchuk’s group Definition The Markov operator
Definition Examples Spectrum of Mξ
Examples Construction Spectral measure
More examples Spectrum of MG
Space of rooted graphs Final remarks
Limit spaces
1
Spinal groups
Spinal groups - Grigorchuk’s group
a
b c d
1
1|1 1|1
1|1
0|0 0|0 0|0
0|1, 1|0
a
b
a
a
1
c
a
1
a
d
1
a
a
Grigorchuk’s group: G = 〈a, b, c , d〉 ≤ Aut(X ∗), X = {0, 1}.
A = 〈a〉 = Z/2Z B = 〈b, c , d〉 = (Z/2Z)2
2
Spinal groups - Definition
We want to generalize Grigorchuk’s group in several ways:
- Action on any regular rooted tree:
d ≥ 2 −→ A = 〈a〉 = Z/dZ
- More elements in B:
m ≥ 1 −→ B = (Z/dZ)m
3
Spinal groups - Definition
Let d ≥ 2 and X = {0, 1, . . . , d − 1}.
Let m ≥ 1, A = Z/dZ = 〈a〉 and B = (Z/dZ)m.
Definition [Bartholdi, Sunic, 2000]
An automaton with states A∪B and alphabet X defines a spinal
group if its edges are of these types
ak 1
b 1
b ω(b)
b ρ(b)
i|i + k
i|i
i 6= 0, d − 1
0|0
d − 1|d − 1
for some epimorphism ω : B → A and automorphism ρ : B → B.
G = 〈A ∪ B〉 ≤ Aut(X ∗)
(The definition can be generalized so that, for every d and m, we obtain an
uncountable family of groups)4
Spinal groups - Examples
B
A
1
b ρ(b)
ω(b) ω(ρ(b))
... ...d − 1|d − 1
...
0|0 0|0
i|i i|i
i|i + j i|i + ka
b
ω(b) 1
ωρ(b) 1
ωρ2(b) 1
5
Spinal groups - Examples
Infinite dihedral
a
b
1
1|1
0|0
0|11|0
D∞ = 〈a, b〉d = 2 ω ρ
m = 1 b 7→ a b 7→ b
The Fabrykowski-Gupta group
1a a2
b b2
2|2 2|2
1|1 1|10|0 0|0
0|11|22|0
0|21|02|1
G = 〈a, a2, b, b2〉ω ρ
d = 3 b 7→ a b 7→ b
m = 1 b2 7→ a2 b2 7→ b2
6
Schreier graphs
Schreier graphs - Definition
Definition
Let G be a group, finitely generated by S = S−1, acting on a set
Y . We define its Schreier graph Sch(G ,S ,Y ) as the graph
given by
• V = Y .
• E = {(z , sz) | z ∈ Y , s ∈ S}.
The graph is oriented and edge-labeled by the set S .
For spinal groups, we will always consider S = (A ∪ B) \ {1}.
7
Schreier graphs - Examples
Grigorchuk’s group: G = 〈a, b, c , d〉
d = 2 m = 2
ω
b 7→ a
c 7→ a
d 7→ 1
ρ
b 7→ c
c 7→ d
d 7→ b
c
b
d
a b
c
d d
a b
d
c c
a b
c
d d
ac
b
d
110 010 000 100 101 001 011 111
Γ3 = Sch(G ,S ,X 3)
8
Schreier graphs - Examples
The Fabrykowski-Gupta group: G = 〈a, a2, b, b2〉
d = 3
m = 1
ω
b 7→ a
b2 7→ a2
ρ
b 7→ b
b2 7→ b2
bbb
b2
b2b2
201
bbb
b2
b2b2
021b
b2121
bb2
221
011
b
b2
111b
b2211
001
bb2
101
200
bbb
b2
b2
b2
020
b
b2
120b
b2220
010
bb2
110
b
b2
210
000b
b2100
202
bbb
b2b2
b2
022
bb2
122
b
b2
222
012b
b2112
bb2
212
002
b
b2
102
Γ3 = Sch(G ,S ,X 3)
9
Schreier graphs - Construction
There is a natural recursive way of constructing Sch(G ,S ,X n+1)
from Sch(G ,S ,X n) (similar to Bondarenko’s inflation of graphs):
Xn
(d-1)n-10
Xn
(d-1)n-10
. . . Xn
(d-1)n-10
Take d copies of Sch(G , S, Xn)
Xn0
(d-1)n-100
Xn1
(d-1)n-101
. . . Xn(d-1)
(d-1)n-10(d-1)
Label each copy accordingly
. . .
Xn0
(d-1)n-100
Xn(d-1)
(d-1)n-10(d-1)Xn1
(d-1)n-101
Connect the vertices (d-1)n-10i
following ωρn−1(b)
∀v0 . . . vn−1 ∈ X n \ {(d − 1)n−10}, ∀i ∈ X , ∀s ∈ S ,
s(v0 . . . vn−1i) = s(v0 . . . vn−1)i
10
Schreier graphs - Construction
The action of G can be extended naturally to the boundary XN of
the tree. Orbits are cofinality classes.
For ξ ∈ XN, the marked graph (Sch(G , S ,Gξ), ξ) is the limit of
(Sch(G ,S ,X n), ξ0 . . . , ξn−1) in the space of rooted graphs.
Definition
A sequence of rooted graphs (Γn, vn) converges to (Γ, v) if
∀r ∈ N, ∃N ∈ N, ∀n ≥ N, Bvn(r) ∼= Bv (r).
11
Schreier graphs - More examples
Grigorchuk’s group: G = 〈a, b, c , d〉
d = 2 m = 2
ω
b 7→ a
c 7→ a
d 7→ 1
ρ
b 7→ c
c 7→ d
d 7→ b
c
b
d
a b
c
d d
a b
d
c c
a b
c
d d
a c
d
b b
a
1111... 0111... 0011... 1011... 1001... 0001... 0101... 1101... 1100...
Sch(G , S , 1N)
a b
c
d d
a b
d
c c
a b
c
d d
a c
d
b b
a b
c
d d
a
0110... 0010... 1010... 1000... 0000... 0100... 1100... 1101... 0101... 0001...
Sch(G , S, 0N)
12
Schreier graphs - More examples
The Fabrykowski-Gupta group: G = 〈a, a2, b, b2〉
d = 3
m = 1
ω
b 7→ a
b2 7→ a2
ρ
b 7→ b
b2 7→ b2
2N
1...
0...
Γ1
01...
21...
Γ1
00... 20...
Γ2
201...
221...
Γ2
200... 220...
Γ3
2201...
2221...
Γ3
2200...
Sch(G , S , 2N)
0N
1...
2...
Γ2
201...
221...
Γ2
202... 222...
Γ4
22201...
22221...
Γ4
22202...
Γ1
01...
21...
Γ1
02...22...
Γ3
2201...
2221...
Γ3
2202...
Sch(G , S, 0N)
13
Schreier graphs - Space of rooted graphs
Let GS ,∗ be the space of rooted graphs with edge labels in S . We
consider the map
F : XN → GS ,∗ξ 7→ (Γξ, ξ)
Remarks
- F is injective.
- F is continuous everywhere except in the orbit of (d − 1)N.
- F(XN) contains only one and two-ended graphs, but F(XN)
contains d-ended graphs as well.
- F(XN) has isolated points iff d = 2.
- The growth of Γξ is polynomial of degree log2(d) [Bondarenko].
14
Schreier graphs - Limit spaces
Nekrashevych defined a notion of limit space JG for a contracting
(finite nucleus) automata group G .
We can embed the graphs Γn in the plane in a way that they
approximate JG :
15
Spectral properties
Spectral properties - The Markov operator
Definition
Let Γ = (V ,E ) be a k-regular graph.
The Markov operator M : `2(V )→ `2(V ) is defined by
Mf (v) =1
k
∑w∼v
f (w)
In our case:
Γn = Sch(G ,S ,X n) −→Mn : `2(X n)→ `2(X n)
Mnf (w) =1
|S |∑s∈S
f (s−1w)
Γξ = Sch(G , S ,Gξ) −→Mξ : `2(Gξ)→ `2(Gξ)
Mξf (η) =1
|S |∑s∈S
f (s−1η)
16
Spectral properties - Spectrum of Mξ
We can exploit the self-similar nature of spinal groups in order to
compute the spectrum of the Markov operator M on Γξ.
Theorem [Dixmier ’77, Proposition 3.4.9]
spec(Mξ) ⊂⋃n≥0
spec(Mn)
Γξ amenable ⇒ spec(Mξ) =⋃n≥0
spec(Mn)
Notice: spec(Mξ) does not depend on ξ.
Bartholdi and Grigorchuk computed the spectrum for Grigorchuk’s
group (two intervals), the Fabrykowski-Gupta group (a Cantor set
plus a countable set), and other related examples.17
Spectral properties - Spectrum of Mξ
We have
Mn =1
|S |(An + Bn)
with
An =
0 1 . . . 1 1
1 0 . . . 1 1...
.... . .
......
1 1 . . . 0 1
1 1 . . . 1 0
Bn =
dm−1An−1 + dm−1 − 1
dm − 1
. . .
dm − 1
Bn−1
18
Spectral properties - Spectrum of Mξ
We use the Schur complement method
det
(A B
C D
)= det(A) det(D − CA−1B)
to find a relation between spec(Mn) and spec(Mn−1):
z ∈ spec(Mn(t))⇐⇒ z ′ ∈ spec(Mn−1(t ′))
z ′ = Φ1(t, z), t ′ = Φ2(t, z)
Solving this recurrence allows to find spec(Mn) explicitly.
19
Spectral properties - Spectrum of Mξ
Theorem [Grigorchuk, Nagnibeda, P.]
spec(Mn) = {1, λ0} ∪ ψ−1
(n−2⋃k=0
F−k(0)
)
spec(Mξ) = {1, λ0} ∪ ψ−1
⋃n≥0
F−n(0)
where F (x) = x2 − d(d − 1) and ψ is a quadratic map.
If d = 2, spec(Mξ) = [− 12m−1 , 0] ∪ [1− 1
2m−1 , 1].
If d ≥ 3, spec(Mξ) is a Cantor set plus a countable set of points.
Notice: spec(Mξ) depends only on d and m.
20
Spectral properties - Spectrum of Mξ
spec(Mξ) is obtained as the preimage by the quadratic map ψ of
the Julia set of F (x) = x2 − d(d − 1).
Julia set of F (x) = x2 − 2 Julia set of F (x) = x2 − 6
Mandelbrot set
21
Spectral properties - Spectral measure
The empirical spectral measure ν of {Γn}n is the weak limit of
the counting measures νn on Γn.
Theorem [Grigorchuk, Nagnibeda, P.]
If d = 2, ν is absolutely continuous with respect to the Lebesgue
measure.
If d ≥ 3, ν is concentrated in the set of eigenvalues of Mn.
−1 −12
0 12
1 −1 −12
0 12
1
22
Spectral properties - Spectrum of MG
We may also consider the Markov operator on Cay(G ,S), the
Cayley graph of G :
MG : `2(G )→ `2(G )
MG f (g) =1
|S |∑s∈S
f (s−1g)
Theorem [Hulanicki]
G amenable ⇒ spec(Mξ) ⊂ spec(MG ) for every ξ ∈ XN.
Theorem [Grigorchuk, Dudko; Grigorchuk, Nagnibeda, P.]
If d = 2, spec(Mξ) = spec(MG ) for every ξ ∈ XN.
23
Spectral properties - Final remarks
Remark
If d = 2, Cayley and Schreier graphs have the same spectrum.
Corollary
There are uncountably many groups whose spectrum is the union
of two intervals.
Corollary
There are uncountably many pairwise non quasi-isometric
isospectral groups.
24
Thank you!
24
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