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!

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