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Mealy machines, automaton (semi)groups, decision problems, and - - PowerPoint PPT Presentation

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Mealy machines, automaton (semi)groups, decision problems, and random generation Thibault Godin S eminaire CALIN Paris 13, October 3, 2017 ANR JCJC 12 JS02 012 01 1 / 35 finite Analogue to Dixon theorem [ANALCO16] groups S k S

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Mealy machines, automaton (semi)groups, decision problems, and random generation

Thibault Godin

S´ eminaire CALIN Paris 13, October 3, 2017

ANR JCJC 12 JS02 012 01 1 / 35

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SLIDE 2

Mealy automata

random generation finite groups infinite groups dynamics

  • f

the action Schreier graphs Wang tillings singular points automaton patterns and group properties finiteness infinite Burnside growth

c a b d e 0|1 1|0 0|0 1|1 0|0 1|1 0|0 1|1 0|0 1|1 z x y x = y i|. i|.

Aℓ−1 Aℓ Aℓ+1 3 1 2 3 ξ singular Am q ξ[0]
  • ξ[0]
ξ[1]
  • ξ[1]
  • ξ[ℓ]
  • ξ[ℓ]
ξ[ℓ + 1]
  • ξ[ℓ + 1]
N x y 1|0 0|1 1|1 0|0 y x y 1 1 y y, 0 x, 0 y, 1 x, 1
  • σ
τ

=      Sk × Sk (Ak × Ak) ⋊ (π, π) Ak × Ak

Invertible reversible non-coreversible automata generate infinite non Burn- side groups [LATA’15 w. Klimann and Picantin] Bireversible automata of prime size cannot generate infinite Burnside groups [MFCS’16 w. Klimann] Analogue to Dixon theorem [ANALCO’16] The set of singular points

  • f a contracting automaton

is described by a B¨ uchi automaton [DGKPR’16]

2 / 35

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SLIDE 3

Mealy automata

dynamics

c a b d e 0|1 1|0 0|0 1|1 0|0 1|1 0|0 1|1 0|0 1|1

3 / 35

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SLIDE 4

c a b d e 0|1 1|0 0|0 1|1 0|0 1|1 0|0 1|1 0|0 1|1

Mealy automaton G A = (Q, Σ, δ, ρ)

4 / 35

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c a b d e 0|1 1|0 0|0 1|1 0|0 1|1 0|0 1|1 0|0 1|1

Mealy automaton G A = (Q, Σ, δ, ρ) ρq : Σ → Σ , q ∈ Q

1 d 1 b

4 / 35

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c a b d e 0|1 1|0 0|0 1|1 0|0 1|1 0|0 1|1 0|0 1|1

Mealy automaton G A = (Q, Σ, δ, ρ) ρq : Σ∗ → Σ∗, q ∈ Q

1 d 1

4 / 35

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SLIDE 7

c a b d e 0|1 1|0 0|0 1|1 0|0 1|1 0|0 1|1 0|0 1|1

Mealy automaton G A = (Q, Σ, δ, ρ) ρq : Σ∗ → Σ∗, q ∈ Q

1 d 1 b 1

4 / 35

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c a b d e 0|1 1|0 0|0 1|1 0|0 1|1 0|0 1|1 0|0 1|1

Mealy automaton G A = (Q, Σ, δ, ρ) ρq : Σ∗ → Σ∗, q ∈ Q

1 d 1 b a 1

4 / 35

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c a b d e 0|1 1|0 0|0 1|1 0|0 1|1 0|0 1|1 0|0 1|1

Mealy automaton G A = (Q, Σ, δ, ρ) ρq : Σ∗ → Σ∗, q ∈ Q

1 d 1 b a 1 e 1

4 / 35

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SLIDE 10

c a b d e 0|1 1|0 0|0 1|1 0|0 1|1 0|0 1|1 0|0 1|1

Mealy automaton G A = (Q, Σ, δ, ρ) ρq : Σ∗ → Σ∗, q ∈ Q

1 d 1 b a 1 e e 1

4 / 35

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c a b d e 0|1 1|0 0|0 1|1 0|0 1|1 0|0 1|1 0|0 1|1

Mealy automaton G A = (Q, Σ, δ, ρ) ρq : Σ∗ → Σ∗, q ∈ Q

1 d 1 b a 1 e e 1 1 e

4 / 35

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c a b d e 0|1 1|0 0|0 1|1 0|0 1|1 0|0 1|1 0|0 1|1

Mealy automaton G A = (Q, Σ, δ, ρ) ρq : Σ∗ → Σ∗, q ∈ Q∗

1 d 1 b a 1 e e 1 1 e a

4 / 35

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c a b d e 0|1 1|0 0|0 1|1 0|0 1|1 0|0 1|1 0|0 1|1

Mealy automaton G A = (Q, Σ, δ, ρ) ρq : Σ∗ → Σ∗, q ∈ Q∗

1 d 1 b a 1 e e 1 1 e a e

4 / 35

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c a b d e 0|1 1|0 0|0 1|1 0|0 1|1 0|0 1|1 0|0 1|1

Mealy automaton G A = (Q, Σ, δ, ρ) ρq : Σ∗ → Σ∗, q ∈ Q∗

1 d 1 b a 1 e e 1 1 e a e e 1 e e 1 e

4 / 35

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SLIDE 15

c a b d e 0|1 1|0 0|0 1|1 0|0 1|1 0|0 1|1 0|0 1|1

Mealy automaton G A = (Q, Σ, δ, ρ) ρq : Σ∗ → Σ∗, q ∈ Q∗

1 d 1 b a 1 e e 1 1 e a e e 1 e e 1 e

ρda(10001) = ρa(ρd(10001))

4 / 35

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c a b d e 0|1 1|0 0|0 1|1 0|0 1|1 0|0 1|1 0|0 1|1

Mealy automaton G A = (Q, Σ, δ, ρ) ρq : Σ∗ → Σ∗, q ∈ Q∗

1 d 1 b a 1 e e 1 1 e a e e 1 e e 1 e

ρda(10001) = ρa(ρd(10001)) A := ρq | q ∈ Q∗

4 / 35

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c a b d e 0|1 1|0 0|0 1|1 0|0 1|1 0|0 1|1 0|0 1|1

Mealy automaton G A = (Q, Σ, δ, ρ) ρq : Σ∗ → Σ∗, q ∈ Q∗

1 d 1 b a 1 e e 1 1 e a e e 1 e e 1 e

ρda(10001) = ρa(ρd(10001)) A := ρq | q ∈ Q∗ da is a state of G2

4 / 35

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Growth Cayley Graph: Γ(G, S)

g h s g.s = h, g, h ∈ G, s ∈ S

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Growth Cayley Graph: Γ(G, S) ex : Z2, {a = (0, 1), b = (1, 0)}

g h s g.s = h, g, h ∈ G, s ∈ S

5 / 35

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Growth Cayley Graph: Γ(G, S) ex : Z2, {a = (0, 1), b = (1, 0)}

g h s g.s = h, g, h ∈ G, s ∈ S

(0, 0) (1, 0) (2, 0) (0, 1) (0, 2) (−1, 0) (−2, 0) (0, −1) (0, −2) (1, 1) (1, −1) (−1, 1) (−1, −1) a a a−1 a−1 a a−1 a a−1 b b−1 b b−1 b b−1 b−1 b

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Growth Cayley Graph: Γ(G, S) ex : Z2, {a = (0, 1), b = (1, 0)}

g h s g.s = h, g, h ∈ G, s ∈ S

(0, 0) (1, 0) (2, 0) (0, 1) (0, 2) (−1, 0) (−2, 0) (0, −1) (0, −2) (1, 1) (1, −1) (−1, 1) (−1, −1) a a a−1 a−1 a a−1 a a−1 b b−1 b b−1 b b−1 b−1 b γ(1)

γ(0) = 1 γ(1) = 5

5 / 35

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Growth Cayley Graph: Γ(G, S) ex : Z2, {a = (0, 1), b = (1, 0)}

g h s g.s = h, g, h ∈ G, s ∈ S

(0, 0) (1, 0) (2, 0) (0, 1) (0, 2) (−1, 0) (−2, 0) (0, −1) (0, −2) (1, 1) (1, −1) (−1, 1) (−1, −1) a a a−1 a−1 a a−1 a a−1 b b−1 b b−1 b b−1 b−1 b γ(1) γ(2)

γ(0) = 1 γ(1) = 5 γ(2) = 13

5 / 35

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Growth Cayley Graph: Γ(G, S) ex : Z2, {a = (0, 1), b = (1, 0)}

g h s g.s = h, g, h ∈ G, s ∈ S

(0, 0) (1, 0) (2, 0) (0, 1) (0, 2) (−1, 0) (−2, 0) (0, −1) (0, −2) (1, 1) (1, −1) (−1, 1) (−1, −1) a a a−1 a−1 a a−1 a a−1 b b−1 b b−1 b b−1 b−1 b γ(1) γ(2)

γ(0) = 1 γ(1) = 5 γ(2) = 13 . . . γ(n) = 2n2 + 2n + 1

5 / 35

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Milnor’s Problem

◮ growth bounded: finite groups

6 / 35

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Milnor’s Problem

◮ growth bounded: finite groups ◮ polynomial growth: Zd, Abelian groups

Γ(Z2)

6 / 35

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Milnor’s Problem

◮ growth bounded: finite groups ◮ polynomial growth: Zd, Abelian groups ◮ exponential growth: Fd

Γ(Z2) Γ(F2)

6 / 35

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Milnor’s Problem

◮ growth bounded: finite groups ◮ polynomial growth: Zd, Abelian groups

Milnor’s Problem (1968):

Do groups with growth growth between polynomial and exponential exist?

◮ exponential growth: Fd

Γ(Z2) Γ(?) Γ(F2)

6 / 35

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Milnor’s Problem

◮ growth bounded: finite groups ◮ polynomial growth: Zd, Abelian groups

Milnor’s Problem (1968):

Do groups with growth growth between polynomial and exponential exist? 1983 (Grigorchuk) Yes, automaton-generated example

◮ exponential growth: Fd

Γ(Z2) Γ(?) Γ(F2)

6 / 35

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Milnor’s Problem

◮ growth bounded: finite groups ◮ polynomial growth: Zd, Abelian groups

Milnor’s Problem (1968):

Do groups with growth growth between polynomial and exponential exist? 1983 (Grigorchuk) Yes, automaton-generated example

◮ exponential growth: Fd

c a b d e 0|1 1|0 0|0 1|1 0|0 1|1 0|0 1|1 0|0 1|1 en0.51 ≤ γ(n) ≤ en0.77

6 / 35

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Order

Order of an element

x ∈ G has finite order if ∃n ≥ 1, xn = e

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Order

Order of an element

x ∈ G has finite order if ∃n ≥ 1, xn = e

◮ Z/nZ : every element has finite order ◮ Z : 0 is the only element of finite order ◮ On the circle R/2πZ : π/2 has finite order, but 1 has infinite order θ = π/2 θ = 1

7 / 35

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The Burnside problem

Burnside (1902):

Can a finitely generated group have all elements of finite order and be infinite?

8 / 35

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The Burnside problem

Burnside (1902):

Can a finitely generated group have all elements of finite order and be infinite? Golod and Shafarevich: yes! (1964)

8 / 35

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The Burnside problem

Burnside (1902):

Can a finitely generated group have all elements of finite order and be infinite? Golod and Shafarevich: yes! (1964) Aleshin+Grigorchuk: an example generated by a Mealy automaton (1972+1980)

8 / 35

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The Burnside problem

Burnside (1902):

Can a finitely generated group have all elements of finite order and be infinite? Golod and Shafarevich: yes! (1964) Aleshin+Grigorchuk: an example generated by a Mealy automaton (1972+1980)

c a b d e 0|1 1|0 0|0 1|1 0|0 1|1 0|0 1|1 0|0 1|1

8 / 35

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SLIDE 36

Mealy automata

random generation finite groups infinite groups dynamics

  • f

the action Schreier graphs Wang tillings singular points automaton patterns and group properties finiteness infinite Burnside growth

c a b d e 0|1 1|0 0|0 1|1 0|0 1|1 0|0 1|1 0|0 1|1 z x y x = y i|. i|.

Aℓ−1 Aℓ Aℓ+1 3 1 2 3 ξ singular Am q ξ[0]
  • ξ[0]
ξ[1]
  • ξ[1]
  • ξ[ℓ]
  • ξ[ℓ]
ξ[ℓ + 1]
  • ξ[ℓ + 1]
N x y 1|0 0|1 1|1 0|0 y x y 1 1 y y, 0 x, 0 y, 1 x, 1
  • σ
τ

=      Sk × Sk (Ak × Ak) ⋊ (π, π) Ak × Ak

Invertible reversible non-coreversible automata generate infinite non Burn- side groups [LATA’15 w. Klimann and Picantin] Bireversible automata of prime size cannot generate infinite Burnside groups [MFCS’16 w. Klimann] Analogue to Dixon theorem [ANALCO’16] The set of singular points

  • f a contracting automaton

is described by a B¨ uchi automaton [DGKPR’16]

9 / 35

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SLIDE 37

automaton patterns and group properties

finiteness infinite Burnside growth

z x y x = y i|. i|.

Aℓ−1 Aℓ Aℓ+1 3 1 2 3

Invertible reversible non-coreversible automata generate infinite non Burn- side groups [LATA’15 w. Klimann and Picantin] Bireversible automata of prime size cannot generate infinite Burnside groups [MFCS’16 w. Klimann] The set of

  • f a contracting

is described automaton

9 / 35

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SLIDE 38

Mealy automata

dynamics

  • f

the action

Schreier graphs Wang tillings singular points

0|1 1|0 0|0 1|1

ξ singular Am q ξ[0]

  • ξ[0]

ξ[1]

  • ξ[1]
  • ξ[ℓ]
  • ξ[ℓ]

ξ[ℓ + 1]

  • ξ[ℓ + 1]

N

x y 1|0 0|1 1|1 0|0

y x y 1 1 y

y, 0 x, 0 y, 1 x, 1 The set of singular points

  • f a contracting automaton

is described by a B¨ uchi automaton [DGKPR’16]

9 / 35

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SLIDE 39

random generation

finite groups infinite groups

  • σ

τ

=      Sk × Sk (Ak × Ak) ⋊ (π, π) Ak × Ak

Analogue to Dixon theorem [ANALCO’16] 9 / 35

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SLIDE 40

Finite random groups

Theorem

Any finite group G is a subgroup of S|G|.

10 / 35

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SLIDE 41

Finite random groups

Theorem

Any finite group G is a subgroup of S|G|.

First idea

Pick up some permutations σ1, . . . , σn of {1, . . . , k}, look at σ1, . . . , σn.

10 / 35

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SLIDE 42

Finite random groups

Theorem

Any finite group G is a subgroup of S|G|.

First idea

Pick up some permutations σ1, . . . , σn of {1, . . . , k}, look at σ1, . . . , σn. # permutations 1 2 3 k!

10 / 35

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SLIDE 43

Finite random groups

Theorem

Any finite group G is a subgroup of S|G|.

First idea

Pick up some permutations σ1, . . . , σn of {1, . . . , k}, look at σ1, . . . , σn. # permutations 1 cyclic groups 2 3 k!

10 / 35

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SLIDE 44

Finite random groups

Theorem

Any finite group G is a subgroup of S|G|.

First idea

Pick up some permutations σ1, . . . , σn of {1, . . . , k}, look at σ1, . . . , σn. # permutations 1 cyclic groups 2 3 k! Sk

10 / 35

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SLIDE 45

Finite random groups

Theorem

Any finite group G is a subgroup of S|G|.

First idea

Pick up some permutations σ1, . . . , σn of {1, . . . , k}, look at σ1, . . . , σn. # permutations 1 cyclic groups ? 2 3 k! Sk

10 / 35

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SLIDE 46

Finite random groups

Theorem (Dixon, 1969)

w.g.p. σ, τ =

  • Sk

Ak # permutations 1 cyclic groups 2 3 k!

10 / 35

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SLIDE 47

Finite random groups

Theorem (Dixon, 1969)

w.g.p. σ, τ =

  • Sk

Ak

Sk

Ak

# permutations 1 cyclic groups 2 3 k!

10 / 35

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SLIDE 48

Finite random groups

Theorem (Dixon, 1969)

w.g.p. σ, τ =

  • Sk

Ak

Sk

Ak

# permutations 1 cyclic groups 2 3 k! Sk or Ak

10 / 35

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SLIDE 49

Random automata

a b c 1|3 2|2 3|1 1|3 3|1 2|2 1|2 2|3 3|1 Is the generated group finite? .

11 / 35

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SLIDE 50

Random automata

a b c 1|3 2|2 3|1 1|3 3|1 2|2 1|2 2|3 3|1 Is the generated group finite? Yes, size 264 · 34.

11 / 35

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SLIDE 51

Random automata

a b c 1|3 2|2 3|1 1|3 3|1 2|2 1|2 2|3 3|1 Is the generated group finite? Yes, size 264 · 34. Difficult problem + unefficient rejection sampling.

11 / 35

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SLIDE 52

Random automata

Antonenko + Russeiev

7 1|1 2|3 3|2 8 1|2 2|3 3|1 9 1|1 2|2 3|3 4 5 3 1 6 2 1|2 2|1 3|3 1|1 2|3 3|2 1|3 3|2 2|1 1|3 3|1 2|2 1|3 3|2 2|1 1|1 2|3 3|2 1|1 2|2 3|3

11 / 35

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SLIDE 53

Random automata

Antonenko + Russeiev cyclic automata

7 1|1 2|3 3|2 8 1|2 2|3 3|1 9 1|1 2|2 3|3 4 5 3 1 6 2 1|2 2|1 3|3 1|1 2|3 3|2 1|3 3|2 2|1 1|3 3|1 2|2 1|3 3|2 2|1 1|1 2|3 3|2 1|1 2|2 3|3

11 / 35

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SLIDE 54

Random 2-state cyclic automata

  • σ

τ

= (σ, τ), (τ, σ)

12 / 35

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SLIDE 55

Random 2-state cyclic automata

  • σ

τ

= (σ, τ), (τ, σ)

Contribution

  • σ

τ

=      Sk × Sk (Ak × Ak) ⋊ (π, π) Ak × Ak

12 / 35

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SLIDE 56

Random 2-state cyclic automata

  • σ

τ

= (σ, τ), (τ, σ)

Contribution

  • σ

τ

=      Sk × Sk (Ak × Ak) ⋊ (π, π) Ak × Ak

Sk × Sk

Ak × Ak (Ak × Ak) ⋊ (π, π)

12 / 35

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SLIDE 57

Random 2-state cyclic automata

  • σ

τ

= (σ, τ), (τ, σ)

Contribution

  • σ

τ

=      Sk × Sk (Ak × Ak) ⋊ (π, π) Ak × Ak

Sk × Sk

Ak × Ak (Ak × Ak) ⋊ (π, π)

Sk × Sk

Ak × Ak

(Ak × Ak) ⋊ (π, π)

Sk × Ak

12 / 35

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SLIDE 58

Random automata

Antonenko + Russeiev cyclic automata

7 1|1 2|3 3|2 8 1|2 2|3 3|1 9 1|1 2|2 3|3 4 5 3 1 6 2 1|2 2|1 3|3 1|1 2|3 3|2 1|3 3|2 2|1 1|3 3|1 2|2 1|3 3|2 2|1 1|1 2|3 3|2 1|1 2|2 3|3

13 / 35

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SLIDE 59

”complexity”

1 . . . n σ1 σn

Sk

Ak

Sk × Sk

Sk × Ak Ak × Ak ⋊ (π, π)

σ τ

Dixon like Dixon like (conj.)

structurally finite structurally infinite

?

a b d e f 0|1 1|0 0|0 1|1 0|0 1|1 0|0 1|1 0|1 1|0

decidable finiteness

2-state bireversible automata

finite by construction

md reduction

a b 0|2 1|1 0|1 1|2 2|0 3|3 2|0 3|3

14 / 35

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SLIDE 60

Asymptotics

Theorem (Dixon 1969,2005)

P(σ, τ = S or A) ∼ 1 − 1/k − 1/k2 − 4/k3 − 23/k4 − 171/k5 − · · ·

15 / 35

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SLIDE 61

Asymptotics

Theorem (Dixon 1969,2005)

P(σ, τ = S or A) ∼ 1 − 1/k − 1/k2 − 4/k3 − 23/k4 − 171/k5 − · · · SNC: ∃w(σ, τ), |w(σ, τ)| = |w(τ, σ)|

Lemma

P(|σ| = |τ|) → 0

15 / 35

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SLIDE 62

Asymptotics

Theorem (Dixon 1969,2005)

P(σ, τ = S or A) ∼ 1 − 1/k − 1/k2 − 4/k3 − 23/k4 − 171/k5 − · · · SNC: ∃w(σ, τ), |w(σ, τ)| = |w(τ, σ)|

Lemma

P(|σ| = |τ|) → 0 proof: [Erd˝

  • s, Tur´

an 1967] log |σ| − 1/2 log2 k 1/ √ 3 log3/2 k → N(0, 1)

15 / 35

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SLIDE 63

Asymptotics

Theorem (Dixon 1969,2005)

P(σ, τ = S or A) ∼ 1 − 1/k − 1/k2 − 4/k3 − 23/k4 − 171/k5 − · · · SNC: ∃w(σ, τ), |w(σ, τ)| = |w(τ, σ)|

Lemma

P(|σ| = |τ|) → 0

Conjecture

P(|σ| = |τ|) ∼ C/k2

15 / 35

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SLIDE 64

Asymptotics

Theorem (Dixon 1969,2005)

P(σ, τ = S or A) ∼ 1 − 1/k − 1/k2 − 4/k3 − 23/k4 − 171/k5 − · · · SNC: ∃w(σ, τ), |w(σ, τ)| = |w(τ, σ)|

Lemma

P(|σ| = |τ|) → 0

Conjecture

P(|σ| = |τ|) ∼ C/k2 rmq: P(|σ| = p) ∼ 1 √pkk(1−1/p) exp (−k(1 − 1/p) + k1/p)

15 / 35

slide-65
SLIDE 65

Mealy automata

dynamics

  • f

the action

Schreier graphs Wang tillings singular points

0|1 1|0 0|0 1|1

ξ singular Am q ξ[0]

  • ξ[0]

ξ[1]

  • ξ[1]
  • ξ[ℓ]
  • ξ[ℓ]

ξ[ℓ + 1]

  • ξ[ℓ + 1]

N

x y 1|0 0|1 1|1 0|0

y x y 1 1 y

y, 0 x, 0 y, 1 x, 1 The set of singular points

  • f a contracting automaton

is described by a B¨ uchi automaton [DGKPR’16]

16 / 35

slide-66
SLIDE 66

Stabilisers and singular points

The stabilisers of an infinite point ξ is StabA(ξ) = {g ∈ A | g(ξ) = ξ}

17 / 35

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SLIDE 67

Stabilisers and singular points

The stabilisers of an infinite point ξ is StabA(ξ) = {g ∈ A | g(ξ) = ξ}

ξ[1] ξ[1] q ξ[2] ξ[2]

  • ξ[3]

ξ[3]

  • ξ[4]

ξ[4]

  • ξ[5]

ξ[5]

  • . . .

17 / 35

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SLIDE 68

Stabilisers and singular points

The stabilisers of an infinite point ξ is StabA(ξ) = {g ∈ A | g(ξ) = ξ}

ξ[1] ξ[1] q ξ[2] ξ[2]

  • ξ[3]

ξ[3]

  • ξ[4]

ξ[4]

  • ξ[5]

ξ[5]

  • . . .

Example

c a b d e 0|1 1|0 0|0 1|1 0|0 1|1 0|0 1|1 0|0 1|1

ρe, ρb, ρc, ρd ∈ StabG(1ω) studied by Y. Vorobets

17 / 35

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SLIDE 69

Stabilisers and singular points

The stabilisers of an infinite point ξ is StabA(ξ) = {g ∈ A | g(ξ) = ξ}

ξ[1] ξ[1] q ξ[2] ξ[2]

  • ξ[3]

ξ[3]

  • ξ[4]

ξ[4]

  • ξ[5]

ξ[5]

  • . . .

Example

c a b d e 0|1 1|0 0|0 1|1 0|0 1|1 0|0 1|1 0|0 1|1

ρe, ρb, ρc, ρd ∈ StabG(1ω) studied by Y. Vorobets

Interesting elements

e a 0|0 1|1 2|2 0|1 1|0 2|2

2ω is stabilised by ρa

17 / 35

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SLIDE 70

Stabilisers and singular points

The stabilisers of an infinite point ξ is StabA(ξ) = {g ∈ A | g(ξ) = ξ}

ξ[1] ξ[1] q ξ[2] ξ[2]

  • ξ[3]

ξ[3]

  • ξ[4]

ξ[4]

  • ξ[5]

ξ[5]

  • . . .

Example

c a b d e 0|1 1|0 0|0 1|1 0|0 1|1 0|0 1|1 0|0 1|1

ρe, ρb, ρc, ρd ∈ StabG(1ω) studied by Y. Vorobets

Interesting elements

e a 0|0 1|1 2|2 0|1 1|0 2|2

2ω is stabilised by ρa

Singular points

ξ singular if ∃g stabilizing ξ and avoiding ending in e

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SLIDE 71

Contracting automata

A contracting ⇐ ⇒ ∃ finite N, ∀q, ∀ξ, ∃n, δξ[:n](q) ∈ N

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SLIDE 72

Contracting automata

A contracting ⇐ ⇒ ∃ finite N, ∀q, ∀ξ, ∃n, δξ[:n](q) ∈ N

Aℓ

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SLIDE 73

Contracting automata

A contracting ⇐ ⇒ ∃ finite N, ∀q, ∀ξ, ∃n, δξ[:n](q) ∈ N

Aℓ q

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SLIDE 74

Contracting automata

A contracting ⇐ ⇒ ∃ finite N, ∀q, ∀ξ, ∃n, δξ[:n](q) ∈ N

Aℓ N q δξ[:n](q) ξ

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SLIDE 75

Contracting automata

A contracting ⇐ ⇒ ∃ finite N, ∀q, ∀ξ, ∃n, δξ[:n](q) ∈ N

Aℓ N q δξ[:n](q) δζ[:n′](q) ξ ζ

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SLIDE 76

Contracting automata

A contracting ⇐ ⇒ ∃ finite N, ∀q, ∀ξ, ∃n, δξ[:n](q) ∈ N

Aℓ N q p δξ[:n](q) δζ[:n′](q) ξ ζ

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SLIDE 77

Contracting automata

A contracting ⇐ ⇒ ∃ finite N, ∀q, ∀ξ, ∃n, δξ[:n](q) ∈ N

Aℓ Am N q p δξ[:n](q) δζ[:n′](q) u v ξ ζ

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SLIDE 78

Contracting automata

A contracting ⇐ ⇒ ∃ finite N, ∀q, ∀ξ, ∃n, δξ[:n](q) ∈ N

Aℓ Am N q p δξ[:n](q) δζ[:n′](q) u v ξ ζ e t 0|0 1|1 0|1 1|1

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SLIDE 79

Contracting automata

A contracting ⇐ ⇒ ∃ finite N, ∀q, ∀ξ, ∃n, δξ[:n](q) ∈ N

Aℓ Am N q p δξ[:n](q) δζ[:n′](q) u v ξ ζ ee et t2 0|0 1|1 0|1 1|1 0|1 1|1

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SLIDE 80

Contracting automata

A contracting ⇐ ⇒ ∃ finite N, ∀q, ∀ξ, ∃n, δξ[:n](q) ∈ N

Aℓ Am N q p δξ[:n](q) δζ[:n′](q) u v ξ ζ e t t2 tℓ 0|0 1|1 0|1 1|1 0|1 1|1 0|1 1|1

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SLIDE 81

Contracting automata and singular points

Basilica automaton N b a ba−1 ab−1 a−1 b−1 e 0|1 0|0 0|0 1|0 0|1 1|1 1|0 1|1 0|1 1|0 0|1 1|1, 0|0 1|0

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SLIDE 82

Contracting automata and singular points

a b a

Basilica automaton N b a ba−1 ab−1 a−1 b−1 e 0|1 0|0 0|0 1|0 0|1 1|1 1|0 1|1 0|1 1|0 0|1 1|1, 0|0 1|0

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SLIDE 83

Contracting automata and singular points

a b 1 a b a e

Basilica automaton N b a ba−1 ab−1 a−1 b−1 e 0|1 0|0 0|0 1|0 0|1 1|1 1|0 1|1 0|1 1|0 0|1 1|1, 0|0 1|0

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SLIDE 84

Contracting automata and singular points

a b 1 a b a e a ∈ N e e

Basilica automaton N b a ba−1 ab−1 a−1 b−1 e 0|1 0|0 0|0 1|0 0|1 1|1 1|0 1|1 0|1 1|0 0|1 1|1, 0|0 1|0

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SLIDE 85

Contracting automata and singular points

ξ singular q ξ[0]

  • ξ[0]

ξ[1]

  • ξ[1]
  • ξ[ℓ]
  • ξ[ℓ]

ξ[ℓ + 1]

  • ξ[ℓ + 1]

Basilica automaton N b a ba−1 ab−1 a−1 b−1 e 0|1 0|0 0|0 1|0 0|1 1|1 1|0 1|1 0|1 1|0 0|1 1|1, 0|0 1|0

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SLIDE 86

Contracting automata and singular points

ξ singular

Am

q ξ[0]

  • ξ[0]

ξ[1]

  • ξ[1]
  • ξ[ℓ]
  • ξ[ℓ]

ξ[ℓ + 1]

  • ξ[ℓ + 1]

Basilica automaton N b a ba−1 ab−1 a−1 b−1 e 0|1 0|0 0|0 1|0 0|1 1|1 1|0 1|1 0|1 1|0 0|1 1|1, 0|0 1|0

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SLIDE 87

Contracting automata and singular points

ξ singular

Am

q ξ[0]

  • ξ[0]

ξ[1]

  • ξ[1]
  • ξ[ℓ]
  • ξ[ℓ]

ξ[ℓ + 1]

  • ξ[ℓ + 1]

N Basilica automaton N b a ba−1 ab−1 a−1 b−1 e 0|1 0|0 0|0 1|0 0|1 1|1 1|0 1|1 0|1 1|0 0|1 1|1, 0|0 1|0

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SLIDE 88

Contracting automata and singular points

ξ singular

Am

q ξ[0]

  • ξ[0]

ξ[1]

  • ξ[1]
  • ξ[ℓ]
  • ξ[ℓ]

ξ[ℓ + 1]

  • ξ[ℓ + 1]

N Basilica automaton N b a ba−1 ab−1 a−1 b−1 e 0|1 0|0 0|0 1|0 0|1 1|1 1|0 1|1 0|1 1|0 0|1 1|1, 0|0 1|0

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SLIDE 89

Contracting automata and singular points

ξ singular

Am

q ξ[0]

  • ξ[0]

ξ[1]

  • ξ[1]
  • ξ[ℓ]
  • ξ[ℓ]

ξ[ℓ + 1]

  • ξ[ℓ + 1]

N Basilica automaton N b a ba−1 ab−1 a−1 b−1 e 0|0 0|0 1|1 1|1 1|1, 0|0

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SLIDE 90

Contracting automata and singular points

ξ singular

Am

q ξ[0]

  • ξ[0]

ξ[1]

  • ξ[1]
  • ξ[ℓ]
  • ξ[ℓ]

ξ[ℓ + 1]

  • ξ[ℓ + 1]

N B¨ uchi automaton b a ba−1 ab−1 a−1 b−1 e 1 1 1, 0

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SLIDE 91

Contracting automata and singular points

ξ singular

Am

q ξ[0]

  • ξ[0]

ξ[1]

  • ξ[1]
  • ξ[ℓ]
  • ξ[ℓ]

ξ[ℓ + 1]

  • ξ[ℓ + 1]

N

Contribution

Sing(B) = ∅.

B¨ uchi automaton b a ba−1 ab−1 a−1 b−1 e 1 1 1, 0

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SLIDE 92

Contracting automata and singular points

ξ singular

Am

q ξ[0]

  • ξ[0]

ξ[1]

  • ξ[1]
  • ξ[ℓ]
  • ξ[ℓ]

ξ[ℓ + 1]

  • ξ[ℓ + 1]

N

c a b d e 0|1 1|0 0|0 1|1 0|0 1|1 0|0 1|1 0|0 1|1

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SLIDE 93

Contracting automata and singular points

ξ singular

Am

q ξ[0]

  • ξ[0]

ξ[1]

  • ξ[1]
  • ξ[ℓ]
  • ξ[ℓ]

ξ[ℓ + 1]

  • ξ[ℓ + 1]

N

c a b d e 0|1 1|0 0|0 1|1 0|0 1|1 0|0 1|1 0|0 1|1 c a b d e 1 1 1 1

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SLIDE 94

Contracting automata and singular points

ξ singular

Am

q ξ[0]

  • ξ[0]

ξ[1]

  • ξ[1]
  • ξ[ℓ]
  • ξ[ℓ]

ξ[ℓ + 1]

  • ξ[ℓ + 1]

N

c a b d e 0|1 1|0 0|0 1|1 0|0 1|1 0|0 1|1 0|0 1|1 c a b d e 1 1 1 1

Proposition [Vorobets, DGKPR]

Sing(G) = (0 + 1)∗1ω.

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SLIDE 95

automaton patterns and group properties

finiteness infinite Burnside growth

z x y x = y i|. i|.

Aℓ−1 Aℓ Aℓ+1 3 1 2 3

Invertible reversible non-coreversible automata generate infinite non Burn- side groups [LATA’15 w. Klimann and Picantin] Bireversible automata of prime size cannot generate infinite Burnside groups [MFCS’16 w. Klimann] The set of

  • f a contracting

is described automaton

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SLIDE 96

About the Grigorchuk automaton

c a b d e 0|1 1|0 0|0 1|1 0|0 1|1 0|0 1|1 0|0 1|1

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SLIDE 97

About the Grigorchuk automaton

c a b d e 0|1 1|0 0|0 1|1 0|0 1|1 0|0 1|1 0|0 1|1

Actions of the states on the letters: ρa : 0 → 1 → 0 ρb, ρc, ρd, ρe : 0 → 0; 1 → 1 →permutations

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SLIDE 98

About the Grigorchuk automaton

c a b d e 0|1 1|0 0|0 1|1 0|0 1|1 0|0 1|1 0|0 1|1

Actions of the states on the letters: ρa : 0 → 1 → 0 ρb, ρc, ρd, ρe : 0 → 0; 1 → 1 →permutations →invertible

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SLIDE 99

About the Grigorchuk automaton

c a b d e 0|1 1|0 0|0 1|1 0|0 1|1 0|0 1|1 0|0 1|1

Actions of the states on the letters: ρa : 0 → 1 → 0 ρb, ρc, ρd, ρe : 0 → 0; 1 → 1 →permutations →invertible Action of a letter on the states: δ0 : a, d, e → e; b, c → a →not a permutation

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SLIDE 100

About the Grigorchuk automaton

c a b d e 0|1 1|0 0|0 1|1 0|0 1|1 0|0 1|1 0|0 1|1

Actions of the states on the letters: ρa : 0 → 1 → 0 ρb, ρc, ρd, ρe : 0 → 0; 1 → 1 →permutations →invertible Action of a letter on the states: δ0 : a, d, e → e; b, c → a →not a permutation →non-reversible

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SLIDE 101

About the Grigorchuk automaton

Reversibility:

Each input letter permutes the stateset.

z x y x = y i|· i|· Actions of the states on the letters: ρa : 0 → 1 → 0 ρb, ρc, ρd, ρe : 0 → 0; 1 → 1 →permutations →invertible Action of a letter on the states: δ0 : a, d, e → e; b, c → a →not a permutation →non-reversible

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SLIDE 102

About the Grigorchuk automaton

Reversibility:

Each input letter permutes the stateset.

z x y x = y i|· i|· Actions of the states on the letters: ρa : 0 → 1 → 0 ρb, ρc, ρd, ρe : 0 → 0; 1 → 1 →permutations →invertible Action of a letter on the states: δ0 : a, d, e → e; b, c → a →not a permutation →non-reversible

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SLIDE 103

Observation

Every known automaton generating an infinite Burnside group happens to be non-reversible.

c a b d e 0|1 1|0 0|0 1|1 0|0 1|1 0|0 1|1 0|0 1|1

b a a−1 e | 1 , 1 | 2 . . . p | 1 0|p, 1|0 . . . p|p − 1 1|1 0|0 2|2, 3|3 . . . p − 1|p − 1 p|p 0|0 1|1 . . . p|p

Question

Can a reversible automaton generate an infinite Burnside group?

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SLIDE 104

Theorem(s)

An invertible and reversible automata which is:

cannot generate an infinite Burnside group.

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SLIDE 105

Theorem(s)

An invertible and reversible automata which is:

2-state

[Klimann] STACS’13

cannot generate an infinite Burnside group.

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SLIDE 106

Theorem(s)

An invertible and reversible automata which is:

2-state connected 3-state

[Klimann] [Klimann, Picantin, and Savchuk] STACS’13 DLT’15

cannot generate an infinite Burnside group.

2 1 2 2 1 1 2 2 2 1 2 2 2 2 2 4 4 4 4 2 1 2 1 4 2 1 2 1 4 4 2 2 1 4 2 2 2 2 2 2 2 2 2 2 2 1 2 2 2 2 4 1 1 1 1 2 2 2 1 1 2 1 4 4 4 4 2 1 1 4 4 2 2 1 2 4 2 1 2 4 2 2 4 2 4 1 4 4 2 1 1 2 4 4 2 1 1 2 2 1 2 1 2 2 2 2 6 2 4 4 2 2 4 2 1 1 2 1 1 1 1 2 2 1 4 1 2

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SLIDE 107

Theorem(s)

An invertible and reversible automata which is:

2-state connected 3-state non coreversible

[Klimann] [Klimann, Picantin, and Savchuk] [G., Klimann, and Picantin] STACS’13 DLT’15 LATA’15

cannot generate an infinite Burnside group.

2 1 2 2 1 1 2 2 2 1 2 2 2 2 2 4 4 4 4 2 1 2 1 4 2 1 2 1 4 4 2 2 1 4 2 2 2 2 2 2 2 2 2 2 2 1 2 2 2 2 4 1 1 1 1 2 2 2 1 1 2 1 4 4 4 4 2 1 1 4 4 2 2 1 2 4 2 1 2 4 2 2 4 2 4 1 4 4 2 1 1 2 4 4 2 1 1 2 2 1 2 1 2 2 2 2 6 2 4 4 2 2 4 2 1 1 2 1 1 1 1 2 2 1 4 1 2

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SLIDE 108

Theorem(s)

An invertible and reversible automata which is:

2-state connected 3-state non coreversible connected with prime size

[Klimann] [Klimann, Picantin, and Savchuk] [G., Klimann, and Picantin] [G. and Klimann] STACS’13 DLT’15 LATA’15 MFCS’16

cannot generate an infinite Burnside group.

2 1 2 2 1 1 2 2 2 1 2 2 2 2 2 4 4 4 4 2 1 2 1 4 2 1 2 1 4 4 2 2 1 4 2 2 2 2 2 2 2 2 2 2 2 1 2 2 2 2 4 1 1 1 1 2 2 2 1 1 2 1 4 4 4 4 2 1 1 4 4 2 2 1 2 4 2 1 2 4 2 2 4 2 4 1 4 4 2 1 1 2 4 4 2 1 1 2 2 1 2 1 2 2 2 2 6 2 4 4 2 2 4 2 1 1 2 1 1 1 1 2 2 1 4 1 2

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SLIDE 109

Schreier tree

c a b 1|1 0|1, 1|0 1|1 0|0 0|0

A

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SLIDE 110

Schreier tree

c a b 1|1 0|1, 1|0 1|1 0|0 0|0 cc aa bb 1|1 0|0, 1|1 1|1 0|0 0|0

ab

ca ac cb ba bc 1|1 0|0 0|1 1|0 0|1 1|0 0|1 1|0 0|0 1|1 1|0 0|1

A A2

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slide-111
SLIDE 111

Schreier tree

c a b 1|1 0|1, 1|0 1|1 0|0 0|0 cc aa bb 1|1 0|0, 1|1 1|1 0|0 0|0

ab

ca ac cb ba bc 1|1 0|0 0|1 1|0 0|1 1|0 0|1 1|0 0|0 1|1 1|0 0|1

A A2

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SLIDE 112

Schreier tree

c a b 1|1 0|1, 1|0 1|1 0|0 0|0 cc aa bb 1|1 0|0, 1|1 1|1 0|0 0|0

ab

ca ac cb ba bc 1|1 0|0 0|1 1|0 0|1 1|0 0|1 1|0 0|0 1|1 1|0 0|1

A A2 1 2

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SLIDE 113

Schreier tree

a aa ab aaa aba aab abb 3 2 1 2 1 1 2 A0 A A2 A3 . . .

size ratio The connected component of A2 containing ab

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SLIDE 114

Boundedness

Proposition

A is finite iff the labels of the cc

  • f (An)n are ultimately 1.

aaaac aaaab aaaaa aaacb aaaca aaabc aaabb aaacc aaaba a aa aaab aaa aaac aaaa 1 1 3 1 1 1 1 1 1 3 1 3 1 1 1 26 / 35

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SLIDE 115

Boundedness

Proposition

A is finite iff the labels of the cc

  • f (An)n are ultimately 1.

Proposition

ρq has finite order iff the labels of the cc containing qn are ultimately 1.

aaaac aaaab aaaaa aaacb aaaca aaabc aaabb aaacc aaaba a aa aaab aaa aaac aaaa 1 1 3 1 1 1 1 1 1 3 1 3 1 1 1 26 / 35

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SLIDE 116

Liftable

Proposition

e liftable to f ⇒ label(e) ≤ label(f ).

aaaa abcb abae

a

abad aa

ba

abca aaab aaa aaba

aba

aab

baba

abc aabc abaa 6 4 2 4 2 4 2 2 4 1 2 4 2 2 4 1 2

vx v C uv uvx, uvy D e f

liftable

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SLIDE 117

Liftable

Proposition

e liftable to f ⇒ label(e) ≤ label(f ).

aaaa abcb abae

a

abad aa

ba

abca aaab aaa aaba

aba

aab

baba

abc aabc abaa 6 4 2 4 2 4 2 2 4 1 2 4 2 2 4 1 2

vx v C uv uvx, uvy D2 e f

liftable

i v v j x k y u h u

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SLIDE 118

Liftable

Proposition

e liftable to f ⇒ label(e) ≤ label(f ).

aaaa abcb abae

a

abad aa

ba

abca aaab aaa aaba

aba

aab

baba

abc aabc abaa 6 4 2 4 2 4 2 2 4 1 2 4 2 2 4 1 2

vx, vy v C uv uvx, uvy D2 e ≥ 2 f

liftable

i v v j x k y u h u

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SLIDE 119

Liftable

Proposition

e liftable to f ⇒ label(e) ≤ label(f ).

aaaa abcb abae

a

abad aa

ba

abca aaab aaa aaba

aba

aab

baba

abc aabc abaa 6 4 2 4 2 4 2 2 4 1 2 4 2 2 4 1 2

vz,vx, vy v C uv uvx, uvy D2 e ≥ 2 f

liftable

i v v j x k y u h u h’ u′ i′ v j′ z k′

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SLIDE 120

Liftable paths

aaaa abcb abae

a

abad aa

ba

abca aaab aaa aaba

aba

aab

baba

abc aabc abaa 6 4 2 4 2 4 2 2 4 1 2 4 2 2 4 1 2 28 / 35

slide-121
SLIDE 121

Liftable paths

aaaa abcb abae

a

abad aa

ba

abca aaab aaa aaba

aba

aab

baba

abc aabc abaa 6 4 2 4 2 4 2 2 4 1 2 4 2 2 4 1 2 28 / 35

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SLIDE 122

Liftable paths

aaaa abcb abae

a

abad aa

ba

abca aaab aaa aaba

aba

aab

baba

abc aabc abaa 6 4 2 4 2 4 2 2 4 1 2 4 2 2 4 1 2 28 / 35

slide-123
SLIDE 123

Liftable paths

aaaa abcb abae

a

abad aa

ba

abca aaab aaa aaba

aba

aab

baba

abc aabc abaa 6 4 2 4 2 4 2 2 4 1 2 4 2 2 4 1 2 28 / 35

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SLIDE 124

Jungle tree

active ≡ labels not ending with 1ω. If active liftable path : not Burnside. Otherwise :

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SLIDE 125

Jungle tree

active ≡ labels not ending with 1ω. If active liftable path : not Burnside. Otherwise :

a aa aaa aaaa aab aaab aabc aabd

7 2 1 1 1 1 1 1

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SLIDE 126

Jungle tree

active ≡ labels not ending with 1ω. If active liftable path : not Burnside. Otherwise : words in the jungle tree have bounded order.

  • nly 1’s

a aa aaa aaaa aab aaab aabc aabd

7 2 1 1 1 1 1 1

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SLIDE 127

3-state case

Aℓ−1 Aℓ Aℓ+1 3 1 1 1 3

Aℓ−1 Aℓ Aℓ+1 3 1 2 3

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SLIDE 128

3-state case

Aℓ−1 Aℓ Aℓ+1 3 1 1 1 3 1 1 1 1 1 1 1 1 1

1

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SLIDE 129

3-state case

Aℓ−1 Aℓ Aℓ+1 3 1 1 1 3 1 1 1 1 1 1 1 1 1

1

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SLIDE 130

3-state case

Aℓ−1 Aℓ Aℓ+1 3 1 1 1 3 1 1 1 1 1 1 1 1 1

Aℓ−1 Aℓ Aℓ+1 3 1 2 3

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SLIDE 131

3-state case

−1

Aℓ Aℓ+1 1 1 1 1 1

Aℓ−1 Aℓ Aℓ+1 3 1 2 Aℓ+2 1 2 30 / 35

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SLIDE 132

3-state case

−1

Aℓ Aℓ+1 1 1 1 1 1

Aℓ−1 Aℓ Aℓ+1 3 1 2 Aℓ+2 1 2 2 1 1 1 1 1 1 1 30 / 35

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SLIDE 133

3-state case

−1

Aℓ Aℓ+1 1 1 1 1 1

Jungle tree Aℓ−1 Aℓ Aℓ+1 3 1 2 Aℓ+2 1 2 2 1 1 1 1 1 1 1 30 / 35

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SLIDE 134

3-state case

Aℓ−1 Aℓ Aℓ+1 3 1 1 1 3 1 1 1 1 1 1 1 1 1

Every word in the tree

Jungle tree Aℓ−1 Aℓ Aℓ+1 3 1 2 3 Aℓ+2 1 2 2 1 1 1 1 1 1 1

not every word in the tree

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SLIDE 135

Looking for (equivalent) words

  • nly 1’s
  • 7

2

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SLIDE 136

Looking for (equivalent) words

  • u

ux0 uy Idea: ∀x0x1x2 · · · find a word with same action in the jungle tree

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SLIDE 137

Looking for (equivalent) words

  • u

ux0 uy ux0s ux0t Idea: ∀x0x1x2 · · · find a word with same action in the jungle tree

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SLIDE 138

Looking for (equivalent) words

  • u

ux0 uy ux0s ux0t ux0w ux0wx1 ∼ ux0x1 ux0ws Idea: ∀x0x1x2 · · · find a word with same action in the jungle tree w ∼ e

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slide-139
SLIDE 139

Looking for (equivalent) words

  • u

ux0 uy ux0s ux0t ux0w ux0wx1 ∼ ux0x1 ux0ws Idea: ∀x0x1x2 · · · find a word with same action in the jungle tree w ∼ e if size prime

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SLIDE 140

automaton patterns and group properties

finiteness infinite Burnside growth

z x y x = y i|. i|.

Aℓ−1 Aℓ Aℓ+1 3 1 2 3

Invertible reversible non-coreversible automata generate infinite non Burn- side groups [LATA’15 w. Klimann and Picantin] Bireversible automata of prime size cannot generate infinite Burnside groups [MFCS’16 w. Klimann] The set of

  • f a contracting

is described automaton

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slide-141
SLIDE 141

automaton patterns and group properties

finiteness infinite Burnside growth

z x y x = y i|. i|.

Aℓ−1 Aℓ Aℓ+1 3 1 2 3

Invertible reversible non-coreversible automata generate infinite non Burn- side groups [LATA’15 w. Klimann and Picantin] Bireversible automata of prime size cannot generate infinite Burnside groups [MFCS’16 w. Klimann] The set of

  • f a contracting

is described automaton Bireversible automata with an element

  • f infinite order have exponential growth

[Klimann’17+]

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SLIDE 142

Structure properties

Invertibility: Each state permutes the alphabet Reversibility: Each input letter permutes the stateset Coreversibility: Each output letter permutes the stateset

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SLIDE 143

Structure properties

Invertibility: Each state permutes the alphabet x i = j i|k j|k Reversibility: Each input letter permutes the stateset z x y x = y i|. i|. Coreversibility: Each output letter permutes the stateset z x y x = y .|k .|k

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SLIDE 144

Structure properties

Invertibility: Each state permutes the alphabet x i = j i|k j|k Reversibility: Each input letter permutes the stateset z x y x = y i|. i|. Coreversibility: Each output letter permutes the stateset z x y x = y .|k .|k

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SLIDE 145

Structure properties

Invertibility: Each state permutes the alphabet x i = j i|k j|k Reversibility: Each input letter permutes the stateset z x y x = y i|. i|. Coreversibility: Each output letter permutes the stateset z x y x = y .|k .|k

Question

How to enumerate or/and (randomly) generate bireversible Mealy automata?

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SLIDE 146

automaton patterns and group properties

finiteness infinite Burnside growth

z x y x = y i|. i|.

Aℓ−1 Aℓ Aℓ+1 3 1 2 3

Invertible reversible non-coreversible automata generate infinite non Burn- side groups [LATA’15 w. Klimann and Picantin] Bireversible automata of prime size cannot generate infinite Burnside groups [MFCS’16 w. Klimann] The set of

  • f a contracting

is described automaton Bireversible automata with an element

  • f infinite order have exponential growth

[Klimann’17+]

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SLIDE 147

Mealy automata

dynamics

  • f

the action

Schreier graphs Wang tillings singular points

0|1 1|0 0|0 1|1

ξ singular Am q ξ[0]

  • ξ[0]

ξ[1]

  • ξ[1]
  • ξ[ℓ]
  • ξ[ℓ]

ξ[ℓ + 1]

  • ξ[ℓ + 1]

N

x y 1|0 0|1 1|1 0|0

y x y 1 1 y

y, 0 x, 0 y, 1 x, 1 The set of singular points

  • f a contracting automaton

is described by a B¨ uchi automaton [DGKPR’16]

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SLIDE 148

random generation

finite groups infinite groups

  • σ

τ

=      Sk × Sk (Ak × Ak) ⋊ (π, π) Ak × Ak

Analogue to Dixon theorem [ANALCO’16] 34 / 35

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SLIDE 149

Thanks!

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