Words in non-periodic branch groups Introduction Groups acting on - - PowerPoint PPT Presentation

SLIDE 1

Words in non-periodic branch groups Elisabeth Fink Introduction Groups acting on Rooted Trees

Rooted Trees Automorphisms

A Construction Construction Words

An Example Lemmata and a Theorem

Growth Further Questions

Words in non-periodic branch groups

Elisabeth Fink

University of Oxford

May 28, 2013

Elisabeth Fink Words in non-periodic branch groups

SLIDE 2

Words in non-periodic branch groups Elisabeth Fink Introduction Groups acting on Rooted Trees

Rooted Trees Automorphisms

A Construction Construction Words

An Example Lemmata and a Theorem

Growth Further Questions

Introduction

Elisabeth Fink Words in non-periodic branch groups

SLIDE 3

Words in non-periodic branch groups Elisabeth Fink Introduction Groups acting on Rooted Trees

Rooted Trees Automorphisms

A Construction Construction Words

An Example Lemmata and a Theorem

Growth Further Questions

Introduction

A construction of a branch group G with

Elisabeth Fink Words in non-periodic branch groups

SLIDE 4

Words in non-periodic branch groups Elisabeth Fink Introduction Groups acting on Rooted Trees

Rooted Trees Automorphisms

A Construction Construction Words

An Example Lemmata and a Theorem

Growth Further Questions

Introduction

A construction of a branch group G with no non-abelian free subgroups

Elisabeth Fink Words in non-periodic branch groups

SLIDE 5

Words in non-periodic branch groups Elisabeth Fink Introduction Groups acting on Rooted Trees

Rooted Trees Automorphisms

A Construction Construction Words

An Example Lemmata and a Theorem

Growth Further Questions

Introduction

A construction of a branch group G with no non-abelian free subgroups exponential growth

Elisabeth Fink Words in non-periodic branch groups

SLIDE 6

Words in non-periodic branch groups Elisabeth Fink Introduction Groups acting on Rooted Trees

Rooted Trees Automorphisms

A Construction Construction Words

An Example Lemmata and a Theorem

Growth Further Questions

Introduction

A construction of a branch group G with no non-abelian free subgroups exponential growth For any g, h ∈ G we construct wg,h(x, y) ∈ F(x, y) with wg,h(g, h) = 1 ∈ G.

Elisabeth Fink Words in non-periodic branch groups

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Words in non-periodic branch groups Elisabeth Fink Introduction Groups acting on Rooted Trees

Rooted Trees Automorphisms

A Construction Construction Words

An Example Lemmata and a Theorem

Growth Further Questions

Rooted Trees

Elisabeth Fink Words in non-periodic branch groups

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Words in non-periodic branch groups Elisabeth Fink Introduction Groups acting on Rooted Trees

Rooted Trees Automorphisms

A Construction Construction Words

An Example Lemmata and a Theorem

Growth Further Questions

Rooted Trees

Rooted tree: cyclefree graph with vertices V and edges E

Elisabeth Fink Words in non-periodic branch groups

SLIDE 9

Words in non-periodic branch groups Elisabeth Fink Introduction Groups acting on Rooted Trees

Rooted Trees Automorphisms

A Construction Construction Words

An Example Lemmata and a Theorem

Growth Further Questions

Rooted Trees

Rooted tree: cyclefree graph with vertices V and edges E distinguished root: r

Elisabeth Fink Words in non-periodic branch groups

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Words in non-periodic branch groups Elisabeth Fink Introduction Groups acting on Rooted Trees

Rooted Trees Automorphisms

A Construction Construction Words

An Example Lemmata and a Theorem

Growth Further Questions

Rooted Trees

Rooted tree: cyclefree graph with vertices V and edges E distinguished root: r distance function: d(v, w), number of edges in unique path from v to w

Elisabeth Fink Words in non-periodic branch groups

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Words in non-periodic branch groups Elisabeth Fink Introduction Groups acting on Rooted Trees

Rooted Trees Automorphisms

A Construction Construction Words

An Example Lemmata and a Theorem

Growth Further Questions

Rooted Trees

Rooted tree: cyclefree graph with vertices V and edges E distinguished root: r distance function: d(v, w), number of edges in unique path from v to w level n: Ω(n) = {v ∈ V : d(v, r) = n}

Elisabeth Fink Words in non-periodic branch groups

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Words in non-periodic branch groups Elisabeth Fink Introduction Groups acting on Rooted Trees

Rooted Trees Automorphisms

A Construction Construction Words

An Example Lemmata and a Theorem

Growth Further Questions

Rooted Trees

Rooted tree: cyclefree graph with vertices V and edges E distinguished root: r distance function: d(v, w), number of edges in unique path from v to w level n: Ω(n) = {v ∈ V : d(v, r) = n} Tn = (V ′, E′) with V ′ = {v ∈ V : d(v, r) ≤ n}

Elisabeth Fink Words in non-periodic branch groups

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Words in non-periodic branch groups Elisabeth Fink Introduction Groups acting on Rooted Trees

Rooted Trees Automorphisms

A Construction Construction Words

An Example Lemmata and a Theorem

Growth Further Questions

Rooted Trees

Rooted tree: cyclefree graph with vertices V and edges E distinguished root: r distance function: d(v, w), number of edges in unique path from v to w level n: Ω(n) = {v ∈ V : d(v, r) = n} Tn = (V ′, E′) with V ′ = {v ∈ V : d(v, r) ≤ n} E′ = {e ∈ E : e = evw, v, w ∈ V ′}

Elisabeth Fink Words in non-periodic branch groups

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Words in non-periodic branch groups Elisabeth Fink Introduction Groups acting on Rooted Trees

Rooted Trees Automorphisms

A Construction Construction Words

An Example Lemmata and a Theorem

Growth Further Questions

Rooted Trees

Rooted tree: cyclefree graph with vertices V and edges E distinguished root: r distance function: d(v, w), number of edges in unique path from v to w level n: Ω(n) = {v ∈ V : d(v, r) = n} Tn = (V ′, E′) with V ′ = {v ∈ V : d(v, r) ≤ n} E′ = {e ∈ E : e = evw, v, w ∈ V ′} Tv: subtree with root v ∈ V

Elisabeth Fink Words in non-periodic branch groups

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Words in non-periodic branch groups Elisabeth Fink Introduction Groups acting on Rooted Trees

Rooted Trees Automorphisms

A Construction Construction Words

An Example Lemmata and a Theorem

Growth Further Questions

Rooted Trees

Rooted tree: cyclefree graph with vertices V and edges E distinguished root: r distance function: d(v, w), number of edges in unique path from v to w level n: Ω(n) = {v ∈ V : d(v, r) = n} Tn = (V ′, E′) with V ′ = {v ∈ V : d(v, r) ≤ n} E′ = {e ∈ E : e = evw, v, w ∈ V ′} Tv: subtree with root v ∈ V

Elisabeth Fink Words in non-periodic branch groups

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Words in non-periodic branch groups Elisabeth Fink Introduction Groups acting on Rooted Trees

Rooted Trees Automorphisms

A Construction Construction Words

An Example Lemmata and a Theorem

Growth Further Questions

Automorphisms acting on rooted trees

Elisabeth Fink Words in non-periodic branch groups

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Words in non-periodic branch groups Elisabeth Fink Introduction Groups acting on Rooted Trees

Rooted Trees Automorphisms

A Construction Construction Words

An Example Lemmata and a Theorem

Growth Further Questions

Automorphisms acting on rooted trees

acting on vertices, preserve edge incidence and root

Elisabeth Fink Words in non-periodic branch groups

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Words in non-periodic branch groups Elisabeth Fink Introduction Groups acting on Rooted Trees

Rooted Trees Automorphisms

A Construction Construction Words

An Example Lemmata and a Theorem

Growth Further Questions

Automorphisms acting on rooted trees

acting on vertices, preserve edge incidence and root stG(n) = {g ∈ G : g acts trivially on Tn}

Elisabeth Fink Words in non-periodic branch groups

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Words in non-periodic branch groups Elisabeth Fink Introduction Groups acting on Rooted Trees

Rooted Trees Automorphisms

A Construction Construction Words

An Example Lemmata and a Theorem

Growth Further Questions

Automorphisms acting on rooted trees

acting on vertices, preserve edge incidence and root stG(n) = {g ∈ G : g acts trivially on Tn} rstG(v) = {g ∈ G : g acts trivially on T\Tv}

Elisabeth Fink Words in non-periodic branch groups

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Words in non-periodic branch groups Elisabeth Fink Introduction Groups acting on Rooted Trees

Rooted Trees Automorphisms

A Construction Construction Words

An Example Lemmata and a Theorem

Growth Further Questions

Automorphisms acting on rooted trees

acting on vertices, preserve edge incidence and root stG(n) = {g ∈ G : g acts trivially on Tn} rstG(v) = {g ∈ G : g acts trivially on T\Tv}

Elisabeth Fink Words in non-periodic branch groups

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Words in non-periodic branch groups Elisabeth Fink Introduction Groups acting on Rooted Trees

Rooted Trees Automorphisms

A Construction Construction Words

An Example Lemmata and a Theorem

Growth Further Questions

Automorphisms acting on rooted trees

acting on vertices, preserve edge incidence and root stG(n) = {g ∈ G : g acts trivially on Tn} rstG(v) = {g ∈ G : g acts trivially on T\Tv}

Elisabeth Fink Words in non-periodic branch groups

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Words in non-periodic branch groups Elisabeth Fink Introduction Groups acting on Rooted Trees

Rooted Trees Automorphisms

A Construction Construction Words

An Example Lemmata and a Theorem

Growth Further Questions

Automorphisms acting on rooted trees

acting on vertices, preserve edge incidence and root stG(n) = {g ∈ G : g acts trivially on Tn} rstG(v) = {g ∈ G : g acts trivially on T\Tv} rstG(n) = Q

v∈Ω(n) rstG(v) Elisabeth Fink Words in non-periodic branch groups

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Words in non-periodic branch groups Elisabeth Fink Introduction Groups acting on Rooted Trees

Rooted Trees Automorphisms

A Construction Construction Words

An Example Lemmata and a Theorem

Growth Further Questions

Automorphisms acting on rooted trees

acting on vertices, preserve edge incidence and root stG(n) = {g ∈ G : g acts trivially on Tn} rstG(v) = {g ∈ G : g acts trivially on T\Tv} rstG(n) = Q

v∈Ω(n) rstG(v)

Definition

A group G is a branch group if it acts transitively on each level of a tree and each rstG(n) has finite index.

Elisabeth Fink Words in non-periodic branch groups

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Words in non-periodic branch groups Elisabeth Fink Introduction Groups acting on Rooted Trees

Rooted Trees Automorphisms

A Construction Construction Words

An Example Lemmata and a Theorem

Growth Further Questions

A specific construction

Elisabeth Fink Words in non-periodic branch groups

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Words in non-periodic branch groups Elisabeth Fink Introduction Groups acting on Rooted Trees

Rooted Trees Automorphisms

A Construction Construction Words

An Example Lemmata and a Theorem

Growth Further Questions

A specific construction

Finite cyclic groups {Ai}i∈N,

Elisabeth Fink Words in non-periodic branch groups

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Words in non-periodic branch groups Elisabeth Fink Introduction Groups acting on Rooted Trees

Rooted Trees Automorphisms

A Construction Construction Words

An Example Lemmata and a Theorem

Growth Further Questions

A specific construction

Finite cyclic groups {Ai}i∈N, ai = Ai

Elisabeth Fink Words in non-periodic branch groups

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Words in non-periodic branch groups Elisabeth Fink Introduction Groups acting on Rooted Trees

Rooted Trees Automorphisms

A Construction Construction Words

An Example Lemmata and a Theorem

Growth Further Questions

A specific construction

Finite cyclic groups {Ai}i∈N, ai = Ai |Ai| = pi, (pi, pj) = 1 for i = j

Elisabeth Fink Words in non-periodic branch groups

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Words in non-periodic branch groups Elisabeth Fink Introduction Groups acting on Rooted Trees

Rooted Trees Automorphisms

A Construction Construction Words

An Example Lemmata and a Theorem

Growth Further Questions

A specific construction

Finite cyclic groups {Ai}i∈N, ai = Ai |Ai| = pi, (pi, pj) = 1 for i = j defining sequence {pi}

Elisabeth Fink Words in non-periodic branch groups

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Words in non-periodic branch groups Elisabeth Fink Introduction Groups acting on Rooted Trees

Rooted Trees Automorphisms

A Construction Construction Words

An Example Lemmata and a Theorem

Growth Further Questions

A specific construction

Finite cyclic groups {Ai}i∈N, ai = Ai |Ai| = pi, (pi, pj) = 1 for i = j defining sequence {pi}

Figure: A tree with p0 = 3, p1 = 5, p2 = 7.

Elisabeth Fink Words in non-periodic branch groups

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Words in non-periodic branch groups Elisabeth Fink Introduction Groups acting on Rooted Trees

Rooted Trees Automorphisms

A Construction Construction Words

An Example Lemmata and a Theorem

Growth Further Questions

A specific construction

Finite cyclic groups {Ai}i∈N, ai = Ai |Ai| = pi, (pi, pj) = 1 for i = j defining sequence {pi}

Elisabeth Fink Words in non-periodic branch groups

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Words in non-periodic branch groups Elisabeth Fink Introduction Groups acting on Rooted Trees

Rooted Trees Automorphisms

A Construction Construction Words

An Example Lemmata and a Theorem

Growth Further Questions

A specific construction

Finite cyclic groups {Ai}i∈N, ai = Ai |Ai| = pi, (pi, pj) = 1 for i = j defining sequence {pi}

G = a, b with

rooted automorphism a0

Elisabeth Fink Words in non-periodic branch groups

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Words in non-periodic branch groups Elisabeth Fink Introduction Groups acting on Rooted Trees

Rooted Trees Automorphisms

A Construction Construction Words

An Example Lemmata and a Theorem

Growth Further Questions

A specific construction

Finite cyclic groups {Ai}i∈N, ai = Ai |Ai| = pi, (pi, pj) = 1 for i = j defining sequence {pi}

G = a, b with

rooted automorphism a0, cyclically permutes Ω(1)

Elisabeth Fink Words in non-periodic branch groups

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Words in non-periodic branch groups Elisabeth Fink Introduction Groups acting on Rooted Trees

Rooted Trees Automorphisms

A Construction Construction Words

An Example Lemmata and a Theorem

Growth Further Questions

A specific construction

Finite cyclic groups {Ai}i∈N, ai = Ai |Ai| = pi, (pi, pj) = 1 for i = j defining sequence {pi}

G = a, b with

rooted automorphism a0, cyclically permutes Ω(1) spinal automorphism b

Elisabeth Fink Words in non-periodic branch groups

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Words in non-periodic branch groups Elisabeth Fink Introduction Groups acting on Rooted Trees

Rooted Trees Automorphisms

A Construction Construction Words

An Example Lemmata and a Theorem

Growth Further Questions

A specific construction

Finite cyclic groups {Ai}i∈N, ai = Ai |Ai| = pi, (pi, pj) = 1 for i = j defining sequence {pi}

G = a, b with

rooted automorphism a0, cyclically permutes Ω(1) spinal automorphism b recursively defined as bn = (bn+1, an+1, 1, . . . , 1)1

Elisabeth Fink Words in non-periodic branch groups

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Words in non-periodic branch groups Elisabeth Fink Introduction Groups acting on Rooted Trees

Rooted Trees Automorphisms

A Construction Construction Words

An Example Lemmata and a Theorem

Growth Further Questions

A specific construction

Finite cyclic groups {Ai}i∈N, ai = Ai |Ai| = pi, (pi, pj) = 1 for i = j defining sequence {pi}

G = a, b with

rooted automorphism a0, cyclically permutes Ω(1) spinal automorphism b recursively defined as bn = (bn+1, an+1, 1, . . . , 1)1

a bi

b1 b2 b3 a1 a2 a3 b4 a4

Elisabeth Fink Words in non-periodic branch groups

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Words in non-periodic branch groups Elisabeth Fink Introduction Groups acting on Rooted Trees

Rooted Trees Automorphisms

A Construction Construction Words

An Example Lemmata and a Theorem

Growth Further Questions

Basic properties of G

Elisabeth Fink Words in non-periodic branch groups

SLIDE 37

Words in non-periodic branch groups Elisabeth Fink Introduction Groups acting on Rooted Trees

Rooted Trees Automorphisms

A Construction Construction Words

An Example Lemmata and a Theorem

Growth Further Questions

Basic properties of G

Proposition G acts on Ω(n) as An−1 ≀ · · · ≀ A0

Elisabeth Fink Words in non-periodic branch groups

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Words in non-periodic branch groups Elisabeth Fink Introduction Groups acting on Rooted Trees

Rooted Trees Automorphisms

A Construction Construction Words

An Example Lemmata and a Theorem

Growth Further Questions

Basic properties of G

Proposition G acts on Ω(n) as An−1 ≀ · · · ≀ A0 Gab = Cp0 × C∞, indirectly implies G is not just infinite

Elisabeth Fink Words in non-periodic branch groups

SLIDE 39

Words in non-periodic branch groups Elisabeth Fink Introduction Groups acting on Rooted Trees

Rooted Trees Automorphisms

A Construction Construction Words

An Example Lemmata and a Theorem

Growth Further Questions

Basic properties of G

Proposition G acts on Ω(n) as An−1 ≀ · · · ≀ A0 Gab = Cp0 × C∞, indirectly implies G is not just infinite 1 = N ⊳ G, then N is finitely generated

Elisabeth Fink Words in non-periodic branch groups

SLIDE 40

Words in non-periodic branch groups Elisabeth Fink Introduction Groups acting on Rooted Trees

Rooted Trees Automorphisms

A Construction Construction Words

An Example Lemmata and a Theorem

Growth Further Questions

Basic properties of G

Proposition G acts on Ω(n) as An−1 ≀ · · · ≀ A0 Gab = Cp0 × C∞, indirectly implies G is not just infinite 1 = N ⊳ G, then N is finitely generated N ⊳ G, N non-trivial, then G/N is soluble

Elisabeth Fink Words in non-periodic branch groups

SLIDE 41

Words in non-periodic branch groups Elisabeth Fink Introduction Groups acting on Rooted Trees

Rooted Trees Automorphisms

A Construction Construction Words

An Example Lemmata and a Theorem

Growth Further Questions

Basic properties of G

Proposition G acts on Ω(n) as An−1 ≀ · · · ≀ A0 Gab = Cp0 × C∞, indirectly implies G is not just infinite 1 = N ⊳ G, then N is finitely generated N ⊳ G, N non-trivial, then G/N is soluble G has infinite virtual first betti number

Elisabeth Fink Words in non-periodic branch groups

SLIDE 42

Words in non-periodic branch groups Elisabeth Fink Introduction Groups acting on Rooted Trees

Rooted Trees Automorphisms

A Construction Construction Words

An Example Lemmata and a Theorem

Growth Further Questions

Basic properties of G

Proposition G acts on Ω(n) as An−1 ≀ · · · ≀ A0 Gab = Cp0 × C∞, indirectly implies G is not just infinite 1 = N ⊳ G, then N is finitely generated N ⊳ G, N non-trivial, then G/N is soluble G has infinite virtual first betti number G is not large

Elisabeth Fink Words in non-periodic branch groups

SLIDE 43

Words in non-periodic branch groups Elisabeth Fink Introduction Groups acting on Rooted Trees

Rooted Trees Automorphisms

A Construction Construction Words

An Example Lemmata and a Theorem

Growth Further Questions

Basic properties of G

Proposition G acts on Ω(n) as An−1 ≀ · · · ≀ A0 Gab = Cp0 × C∞, indirectly implies G is not just infinite 1 = N ⊳ G, then N is finitely generated N ⊳ G, N non-trivial, then G/N is soluble G has infinite virtual first betti number G is not large G(n+1) ≤ rstG(n)

Elisabeth Fink Words in non-periodic branch groups

SLIDE 44

Words in non-periodic branch groups Elisabeth Fink Introduction Groups acting on Rooted Trees

Rooted Trees Automorphisms

A Construction Construction Words

An Example Lemmata and a Theorem

Growth Further Questions

An Example

g1 = (ab)4a−4 = (b1, b1a1, b1a1, b1a1, a1, 1, . . . , 1)1,

Elisabeth Fink Words in non-periodic branch groups

SLIDE 45

Words in non-periodic branch groups Elisabeth Fink Introduction Groups acting on Rooted Trees

Rooted Trees Automorphisms

A Construction Construction Words

An Example Lemmata and a Theorem

Growth Further Questions

An Example

g1 = (ab)4a−4 = (b1, b1a1, b1a1, b1a1, a1, 1, . . . , 1)1, b1 b1a1 b1a1 b1a1 a1 1

Elisabeth Fink Words in non-periodic branch groups

SLIDE 46

Words in non-periodic branch groups Elisabeth Fink Introduction Groups acting on Rooted Trees

Rooted Trees Automorphisms

A Construction Construction Words

An Example Lemmata and a Theorem

Growth Further Questions

An Example

g1 = (ab)4a−4 = (b1, b1a1, b1a1, b1a1, a1, 1, . . . , 1)1, b1 b1a1 b1a1 b1a1 a1 1 g2 = [b, ba] = ` b−1

1 , b1a−1 1 , a1, 1, . . . , 1

´

1, Elisabeth Fink Words in non-periodic branch groups

SLIDE 47

Words in non-periodic branch groups Elisabeth Fink Introduction Groups acting on Rooted Trees

Rooted Trees Automorphisms

A Construction Construction Words

An Example Lemmata and a Theorem

Growth Further Questions

An Example

g1 = (ab)4a−4 = (b1, b1a1, b1a1, b1a1, a1, 1, . . . , 1)1, b1 b1a1 b1a1 b1a1 a1 1 g2 = [b, ba] = ` b−1

1 , b1a−1 1 , a1, 1, . . . , 1

´

1,

b−1

1

b1a−1

1

a1 1

Elisabeth Fink Words in non-periodic branch groups

SLIDE 48

Words in non-periodic branch groups Elisabeth Fink Introduction Groups acting on Rooted Trees

Rooted Trees Automorphisms

A Construction Construction Words

An Example Lemmata and a Theorem

Growth Further Questions

An Example

g1 = (ab)4a−4, g2 = [b, ba]

b1 b1a1 b1a1b1a1 a1 1 b−1

1 b1a−1 1 a1

1

Elisabeth Fink Words in non-periodic branch groups

SLIDE 49

Words in non-periodic branch groups Elisabeth Fink Introduction Groups acting on Rooted Trees

Rooted Trees Automorphisms

A Construction Construction Words

An Example Lemmata and a Theorem

Growth Further Questions

An Example

g1 = (ab)4a−4, g2 = [b, ba]

b1 b1a1 b1a1b1a1 a1 1 b−1

1 b1a−1 1 a1

1

c1 = [g1, g2] =

“ 1, “ a2, b−1

2

, a−1

2

, 1, . . . , b2 ” , “ 1, b−1

2

, a−1

2

b2, a2, 1, . . . , 1 ” , 1, . . . , 1 ”

2

Elisabeth Fink Words in non-periodic branch groups

SLIDE 50

Words in non-periodic branch groups Elisabeth Fink Introduction Groups acting on Rooted Trees

Rooted Trees Automorphisms

A Construction Construction Words

An Example Lemmata and a Theorem

Growth Further Questions

An Example

g1 = (ab)4a−4, g2 = [b, ba]

b1 b1a1 b1a1b1a1 a1 1 b−1

1 b1a−1 1 a1

1

c1 = [g1, g2] =

“ 1, “ a2, b−1

2

, a−1

2

, 1, . . . , b2 ” , “ 1, b−1

2

, a−1

2

b2, a2, 1, . . . , 1 ” , 1, . . . , 1 ”

2

a2 b−1

2

a−1

2

1 b2 1 b2 a−1

2

b2 a2 1 Elisabeth Fink Words in non-periodic branch groups

SLIDE 51

Words in non-periodic branch groups Elisabeth Fink Introduction Groups acting on Rooted Trees

Rooted Trees Automorphisms

A Construction Construction Words

An Example Lemmata and a Theorem

Growth Further Questions

An Example

c1 = [g1, g2]

a2 b−1

2 a−1 2

1 b2 1 b2 a−1

2

b2a2 1 Elisabeth Fink Words in non-periodic branch groups

SLIDE 52

Words in non-periodic branch groups Elisabeth Fink Introduction Groups acting on Rooted Trees

Rooted Trees Automorphisms

A Construction Construction Words

An Example Lemmata and a Theorem

Growth Further Questions

An Example

c1 = [g1, g2]

a2 b−1

2 a−1 2

1 b2 1 b2 a−1

2

b2a2 1

cg4

1

1 = [g1, g2]g4

1

Elisabeth Fink Words in non-periodic branch groups

SLIDE 53

Words in non-periodic branch groups Elisabeth Fink Introduction Groups acting on Rooted Trees

Rooted Trees Automorphisms

A Construction Construction Words

An Example Lemmata and a Theorem

Growth Further Questions

An Example

c1 = [g1, g2]

a2 b−1

2 a−1 2

1 b2 1 b2 a−1

2

b2a2 1

cg4

1

1 = [g1, g2]g4

1 =

“ 1, “ 1, 1, 1, b2, a2, b−1

2

, a−1

2

b−1

2

a−1

2

b2a2, 1, . . . , 1 ” , “ 1, 1, 1, 1, b−1

2

, a−1

2

b−1

2

a−1

2

b2

2a2, a−1 2

b−1

2

a2b2a2, 1, . . . , 1 ” , 1, . . . , 1 ”

2 .

Elisabeth Fink Words in non-periodic branch groups

SLIDE 54

Words in non-periodic branch groups Elisabeth Fink Introduction Groups acting on Rooted Trees

Rooted Trees Automorphisms

A Construction Construction Words

An Example Lemmata and a Theorem

Growth Further Questions

An Example

c1 = [g1, g2]

a2 b−1

2 a−1 2

1 b2 1 b2 a−1

2

b2a2 1

cg4

1

1 = [g1, g2]g4

1 =

“ 1, “ 1, 1, 1, b2, a2, b−1

2

, a−1

2

b−1

2

a−1

2

b2a2, 1, . . . , 1 ” , “ 1, 1, 1, 1, b−1

2

, a−1

2

b−1

2

a−1

2

b2

2a2, a−1 2

b−1

2

a2b2a2, 1, . . . , 1 ” , 1, . . . , 1 ”

2 .

1 1 1 b2 a2 a−1

2

b−1

2

a−1

2

b2a2 1 1 1 1 b−1

2

1 a−1

2

b−1

2

a−1

2

b2

2a2 a−1 2

b−1

2

a2b2a2 1 Elisabeth Fink Words in non-periodic branch groups

SLIDE 55

Words in non-periodic branch groups Elisabeth Fink Introduction Groups acting on Rooted Trees

Rooted Trees Automorphisms

A Construction Construction Words

An Example Lemmata and a Theorem

Growth Further Questions

An Example

c1 = [g1, g2]

a2 b−1

2 a−1 2

1 b2 1 b2 a−1

2

b2a2 1

c

g4

1

1 = [g1, g2]g4

1

1 1 1 b2 a2 a−1

2

b−1

2

a−1

2

b2a2 1 1 1 1 b−1

2

1 a−1

2

b−1

2

a−1

2

b2

2a2 a−1 2

b−1

2

a2b2a2 1 Elisabeth Fink Words in non-periodic branch groups

SLIDE 56

Words in non-periodic branch groups Elisabeth Fink Introduction Groups acting on Rooted Trees

Rooted Trees Automorphisms

A Construction Construction Words

An Example Lemmata and a Theorem

Growth Further Questions

An Example

c1 = [g1, g2]

a2 b−1

2 a−1 2

1 b2 1 b2 a−1

2

b2a2 1

c

g4

1

1 = [g1, g2]g4

1

1 1 1 b2 a2 a−1

2

b−1

2

a−1

2

b2a2 1 1 1 1 b−1

2

1 a−1

2

b−1

2

a−1

2

b2

2a2 a−1 2

b−1

2

a2b2a2 1

Comparing pictures:

• c1, cg4

1

1

• = 1

Elisabeth Fink Words in non-periodic branch groups

SLIDE 57

Words in non-periodic branch groups Elisabeth Fink Introduction Groups acting on Rooted Trees

Rooted Trees Automorphisms

A Construction Construction Words

An Example Lemmata and a Theorem

Growth Further Questions

An Example

c1 = [g1, g2]

a2 b−1

2 a−1 2

1 b2 1 b2 a−1

2

b2a2 1

c

g4

1

1 = [g1, g2]g4

1

1 1 1 b2 a2 a−1

2

b−1

2

a−1

2

b2a2 1 1 1 1 b−1

2

1 a−1

2

b−1

2

a−1

2

b2

2a2 a−1 2

b−1

2

a2b2a2 1

Comparing pictures:

• c1, cg4

1

1

• = 1

wg1,g2(x, y) =

• [x, y], [x, y]x4

with wg1,g2 (g1, g2) = 1

Elisabeth Fink Words in non-periodic branch groups

SLIDE 58

Words in non-periodic branch groups Elisabeth Fink Introduction Groups acting on Rooted Trees

Rooted Trees Automorphisms

A Construction Construction Words

An Example Lemmata and a Theorem

Growth Further Questions

Lemmata

Definition

Let g ∈ G, q ∈ Z. A spine s is a g-conjugate of a power of the generator b: s = g−1bqg.

Elisabeth Fink Words in non-periodic branch groups

SLIDE 59

Words in non-periodic branch groups Elisabeth Fink Introduction Groups acting on Rooted Trees

Rooted Trees Automorphisms

A Construction Construction Words

An Example Lemmata and a Theorem

Growth Further Questions

Lemmata

Definition

Let g ∈ G, q ∈ Z. A spine s is a g-conjugate of a power of the generator b: s = g−1bqg. Number of spines of g: ζ(g) = min n sg : g = ak · Qsg

i=1 a−ki bqi aki

• Elisabeth Fink

Words in non-periodic branch groups

SLIDE 60

Words in non-periodic branch groups Elisabeth Fink Introduction Groups acting on Rooted Trees

Rooted Trees Automorphisms

A Construction Construction Words

An Example Lemmata and a Theorem

Growth Further Questions

Lemmata

Definition

Let g ∈ G, q ∈ Z. A spine s is a g-conjugate of a power of the generator b: s = g−1bqg. Number of spines of g: ζ(g) = min n sg : g = ak · Qsg

i=1 a−ki bqi aki

• Elisabeth Fink

Words in non-periodic branch groups

SLIDE 61

Words in non-periodic branch groups Elisabeth Fink Introduction Groups acting on Rooted Trees

Rooted Trees Automorphisms

A Construction Construction Words

An Example Lemmata and a Theorem

Growth Further Questions

Lemmata

Definition

Let g ∈ G, q ∈ Z. A spine s is a g-conjugate of a power of the generator b: s = g−1bqg. Number of spines of g: ζ(g) = min n sg : g = ak · Qsg

i=1 a−ki bqi aki

• c0 = g, c1 = [g1, g2] ,

Elisabeth Fink Words in non-periodic branch groups

SLIDE 62

Words in non-periodic branch groups Elisabeth Fink Introduction Groups acting on Rooted Trees

Rooted Trees Automorphisms

A Construction Construction Words

An Example Lemmata and a Theorem

Growth Further Questions

Lemmata

Definition

Let g ∈ G, q ∈ Z. A spine s is a g-conjugate of a power of the generator b: s = g−1bqg. Number of spines of g: ζ(g) = min n sg : g = ak · Qsg

i=1 a−ki bqi aki

• c0 = g, c1 = [g1, g2] , ci =

h ci−1, c

ci−2 i−1

i , i ≥ 2

Elisabeth Fink Words in non-periodic branch groups

SLIDE 63

Words in non-periodic branch groups Elisabeth Fink Introduction Groups acting on Rooted Trees

Rooted Trees Automorphisms

A Construction Construction Words

An Example Lemmata and a Theorem

Growth Further Questions

Lemmata

Definition

Let g ∈ G, q ∈ Z. A spine s is a g-conjugate of a power of the generator b: s = g−1bqg. Number of spines of g: ζ(g) = min n sg : g = ak · Qsg

i=1 a−ki bqi aki

• c0 = g, c1 = [g1, g2] , ci =

h ci−1, c

ci−2 i−1

i , i ≥ 2 Lemma ζ (ci) ≤ 5i (ζ (g1) + ζ (g2)) , i ≥ 0

Elisabeth Fink Words in non-periodic branch groups

SLIDE 64

Words in non-periodic branch groups Elisabeth Fink Introduction Groups acting on Rooted Trees

Rooted Trees Automorphisms

A Construction Construction Words

An Example Lemmata and a Theorem

Growth Further Questions

Lemmata

Definition

Let g ∈ G, q ∈ Z. A spine s is a g-conjugate of a power of the generator b: s = g−1bqg. Number of spines of g: ζ(g) = min n sg : g = ak · Qsg

i=1 a−ki bqi aki

• c0 = g, c1 = [g1, g2] , ci =

h ci−1, c

ci−2 i−1

i , i ≥ 2 Lemma ζ (ci) ≤ 5i (ζ (g1) + ζ (g2)) , i ≥ 0 ci ∈ rstG(i − 1) ≤ stG(i − 1), i ≥ 2

Elisabeth Fink Words in non-periodic branch groups

SLIDE 65

Words in non-periodic branch groups Elisabeth Fink Introduction Groups acting on Rooted Trees

Rooted Trees Automorphisms

A Construction Construction Words

An Example Lemmata and a Theorem

Growth Further Questions

Lemmata

Definition

Let g ∈ G, q ∈ Z. A spine s is a g-conjugate of a power of the generator b: s = g−1bqg. Number of spines of g: ζ(g) = min n sg : g = ak · Qsg

i=1 a−ki bqi aki

• c0 = g, c1 = [g1, g2] , ci =

h ci−1, c

ci−2 i−1

i , i ≥ 2 Lemma ζ (ci) ≤ 5i (ζ (g1) + ζ (g2)) , i ≥ 0 ci ∈ rstG(i − 1) ≤ stG(i − 1), i ≥ 2 ci stabilises level i!

Elisabeth Fink Words in non-periodic branch groups

SLIDE 66

Words in non-periodic branch groups Elisabeth Fink Introduction Groups acting on Rooted Trees

Rooted Trees Automorphisms

A Construction Construction Words

An Example Lemmata and a Theorem

Growth Further Questions

Theorem

c0 = g, c1 = [g1, g2] , ci = h ci−1, c

ci−2 i−1

i , i ≥ 2

Elisabeth Fink Words in non-periodic branch groups

SLIDE 67

Words in non-periodic branch groups Elisabeth Fink Introduction Groups acting on Rooted Trees

Rooted Trees Automorphisms

A Construction Construction Words

An Example Lemmata and a Theorem

Growth Further Questions

Theorem

c0 = g, c1 = [g1, g2] , ci = h ci−1, c

ci−2 i−1

i , i ≥ 2 Proposition ci has the form ci = ` di,1, . . . , di,mi ´

i with di,j one of the following

1

di,j = bt

i , t ∈ Z Elisabeth Fink Words in non-periodic branch groups

SLIDE 68

Words in non-periodic branch groups Elisabeth Fink Introduction Groups acting on Rooted Trees

Rooted Trees Automorphisms

A Construction Construction Words

An Example Lemmata and a Theorem

Growth Further Questions

Theorem

c0 = g, c1 = [g1, g2] , ci = h ci−1, c

ci−2 i−1

i , i ≥ 2 Proposition ci has the form ci = ` di,1, . . . , di,mi ´

i with di,j one of the following

1

di,j = bt

i , t ∈ Z

2

di,j = aqb, q = k · pi, b ∈ B ∩ stG(n)

Elisabeth Fink Words in non-periodic branch groups

SLIDE 69

Words in non-periodic branch groups Elisabeth Fink Introduction Groups acting on Rooted Trees

Rooted Trees Automorphisms

A Construction Construction Words

An Example Lemmata and a Theorem

Growth Further Questions

Theorem

c0 = g, c1 = [g1, g2] , ci = h ci−1, c

ci−2 i−1

i , i ≥ 2 Proposition ci has the form ci = ` di,1, . . . , di,mi ´

i with di,j one of the following

1

di,j = bt

i , t ∈ Z

2

di,j = aqb, q = k · pi, b ∈ B ∩ stG(n)

3

recursive: di,j = (di+1,t1, . . . , di+1,tm)i+1

Elisabeth Fink Words in non-periodic branch groups

SLIDE 70

Words in non-periodic branch groups Elisabeth Fink Introduction Groups acting on Rooted Trees

Rooted Trees Automorphisms

A Construction Construction Words

An Example Lemmata and a Theorem

Growth Further Questions

Theorem

c0 = g, c1 = [g1, g2] , ci = h ci−1, c

ci−2 i−1

i , i ≥ 2 Proposition ci has the form ci = ` di,1, . . . , di,mi ´

i with di,j one of the following

1

di,j = bt

i , t ∈ Z

2

di,j = aqb, q = k · pi, b ∈ B ∩ stG(n)

3

recursive: di,j = (di+1,t1, . . . , di+1,tm)i+1

4

di,j = 1

Elisabeth Fink Words in non-periodic branch groups

SLIDE 71

Words in non-periodic branch groups Elisabeth Fink Introduction Groups acting on Rooted Trees

Rooted Trees Automorphisms

A Construction Construction Words

An Example Lemmata and a Theorem

Growth Further Questions

Theorem

c0 = g, c1 = [g1, g2] , ci = h ci−1, c

ci−2 i−1

i , i ≥ 2 Proposition ci has the form ci = ` di,1, . . . , di,mi ´

i with di,j one of the following

1

di,j = bt

i , t ∈ Z

2

di,j = aqb, q = k · pi, b ∈ B ∩ stG(n)

3

recursive: di,j = (di+1,t1, . . . , di+1,tm)i+1

4

di,j = 1 ∃n ∈ N with 3 only occurs for di,j with i ≤ n.

Elisabeth Fink Words in non-periodic branch groups

SLIDE 72

Words in non-periodic branch groups Elisabeth Fink Introduction Groups acting on Rooted Trees

Rooted Trees Automorphisms

A Construction Construction Words

An Example Lemmata and a Theorem

Growth Further Questions

Theorem

c0 = g, c1 = [g1, g2] , ci = h ci−1, c

ci−2 i−1

i , i ≥ 2 Proposition ci has the form ci = ` di,1, . . . , di,mi ´

i with di,j one of the following

1

di,j = bt

i , t ∈ Z

2

di,j = aqb, q = k · pi, b ∈ B ∩ stG(n)

3

recursive: di,j = (di+1,t1, . . . , di+1,tm)i+1

4

di,j = 1 ∃n ∈ N with 3 only occurs for di,j with i ≤ n. Theorem If pi ≥ (25pi−1)3 Qi−1

k=0 pk , then for each g, h ∈ G there exists a word

wg,h(x, y) ∈ F(x, y) with wg,h(g, h) = 1 ∈ G.

Elisabeth Fink Words in non-periodic branch groups

SLIDE 73

Words in non-periodic branch groups Elisabeth Fink Introduction Groups acting on Rooted Trees

Rooted Trees Automorphisms

A Construction Construction Words

An Example Lemmata and a Theorem

Growth Further Questions

Theorem

c0 = g, c1 = [g1, g2] , ci = h ci−1, c

ci−2 i−1

i , i ≥ 2 Proposition ci has the form ci = ` di,1, . . . , di,mi ´

i with di,j one of the following

1

di,j = bt

i , t ∈ Z

2

di,j = aqb, q = k · pi, b ∈ B ∩ stG(n)

3

recursive: di,j = (di+1,t1, . . . , di+1,tm)i+1

4

di,j = 1 ∃n ∈ N with 3 only occurs for di,j with i ≤ n. Theorem If pi ≥ (25pi−1)3 Qi−1

k=0 pk , then for each g, h ∈ G there exists a word

wg,h(x, y) ∈ F(x, y) with wg,h(g, h) = 1 ∈ G. Hence G has no non-abelian free subgroups.

Elisabeth Fink Words in non-periodic branch groups

SLIDE 74

Words in non-periodic branch groups Elisabeth Fink Introduction Groups acting on Rooted Trees

Rooted Trees Automorphisms

A Construction Construction Words

An Example Lemmata and a Theorem

Growth Further Questions

Growth

Elisabeth Fink Words in non-periodic branch groups

SLIDE 75

Words in non-periodic branch groups Elisabeth Fink Introduction Groups acting on Rooted Trees

Rooted Trees Automorphisms

A Construction Construction Words

An Example Lemmata and a Theorem

Growth Further Questions

Growth

Proposition

G does not have polynomial growth.

Elisabeth Fink Words in non-periodic branch groups

SLIDE 76

Words in non-periodic branch groups Elisabeth Fink Introduction Groups acting on Rooted Trees

Rooted Trees Automorphisms

A Construction Construction Words

An Example Lemmata and a Theorem

Growth Further Questions

Growth

Proposition

G does not have polynomial growth.

Self-Similarities

Elisabeth Fink Words in non-periodic branch groups

SLIDE 77

Words in non-periodic branch groups Elisabeth Fink Introduction Groups acting on Rooted Trees

Rooted Trees Automorphisms

A Construction Construction Words

An Example Lemmata and a Theorem

Growth Further Questions

Growth

Proposition

G does not have polynomial growth.

Self-Similarities

Find copies of G1 in G:

Elisabeth Fink Words in non-periodic branch groups

SLIDE 78

Words in non-periodic branch groups Elisabeth Fink Introduction Groups acting on Rooted Trees

Rooted Trees Automorphisms

A Construction Construction Words

An Example Lemmata and a Theorem

Growth Further Questions

Growth

Proposition

G does not have polynomial growth.

Self-Similarities

Find copies of G1 in G: G1 = a1, b1 G1 = a1, b1

b1 a1 a1 b1 b1 b1 a1

Elisabeth Fink Words in non-periodic branch groups

SLIDE 79

Words in non-periodic branch groups Elisabeth Fink Introduction Groups acting on Rooted Trees

Rooted Trees Automorphisms

A Construction Construction Words

An Example Lemmata and a Theorem

Growth Further Questions

Growth

Elisabeth Fink Words in non-periodic branch groups

SLIDE 80

Words in non-periodic branch groups Elisabeth Fink Introduction Groups acting on Rooted Trees

Rooted Trees Automorphisms

A Construction Construction Words

An Example Lemmata and a Theorem

Growth Further Questions

Growth

Lemma γG “Qi

k=1 p2 kn

” ≥ γGi (n)

Qi

k=1 pk /3.

Elisabeth Fink Words in non-periodic branch groups

SLIDE 81

Words in non-periodic branch groups Elisabeth Fink Introduction Groups acting on Rooted Trees

Rooted Trees Automorphisms

A Construction Construction Words

An Example Lemmata and a Theorem

Growth Further Questions

Growth

Lemma γG “Qi

k=1 p2 kn

” ≥ γGi (n)

Qi

k=1 pk /3.

Proposition γBi (2pi) ≥ 2pi −1 · (pi − 1)

pi 2 −2.

Elisabeth Fink Words in non-periodic branch groups

SLIDE 82

Words in non-periodic branch groups Elisabeth Fink Introduction Groups acting on Rooted Trees

Rooted Trees Automorphisms

A Construction Construction Words

An Example Lemmata and a Theorem

Growth Further Questions

Growth

Lemma γG “Qi

k=1 p2 kn

” ≥ γGi (n)

Qi

k=1 pk /3.

Proposition γBi (2pi) ≥ 2pi −1 · (pi − 1)

pi 2 −2.

Theorem If {pi} satisfies log (pi − 1) ≥ 5 · ` 47

5

´i · Qi

k=0 pk for all sufficiently large i,

then G has exponential growth.

Elisabeth Fink Words in non-periodic branch groups

SLIDE 83

Words in non-periodic branch groups Elisabeth Fink Introduction Groups acting on Rooted Trees

Rooted Trees Automorphisms

A Construction Construction Words

An Example Lemmata and a Theorem

Growth Further Questions

Further Questions

Is G amenable?

Elisabeth Fink Words in non-periodic branch groups

SLIDE 84

Words in non-periodic branch groups Elisabeth Fink Introduction Groups acting on Rooted Trees

Rooted Trees Automorphisms

A Construction Construction Words

An Example Lemmata and a Theorem

Growth Further Questions

Further Questions

Is G amenable? Growth in all cases?

Elisabeth Fink Words in non-periodic branch groups

SLIDE 85

Words in non-periodic branch groups Elisabeth Fink Introduction Groups acting on Rooted Trees

Rooted Trees Automorphisms

A Construction Construction Words

An Example Lemmata and a Theorem

Growth Further Questions

Further Questions

Is G amenable? Growth in all cases? Similar examples are amenable for slow growth (Brieussel)

Elisabeth Fink Words in non-periodic branch groups

SLIDE 86

Words in non-periodic branch groups Elisabeth Fink Introduction Groups acting on Rooted Trees

Rooted Trees Automorphisms

A Construction Construction Words

An Example Lemmata and a Theorem

Growth Further Questions

Further Questions

Is G amenable? Growth in all cases? Similar examples are amenable for slow growth (Brieussel) Does G contain a free semigroup?

Elisabeth Fink Words in non-periodic branch groups

SLIDE 87

Words in non-periodic branch groups Elisabeth Fink Introduction Groups acting on Rooted Trees

Rooted Trees Automorphisms

A Construction Construction Words

An Example Lemmata and a Theorem

Growth Further Questions

Thank you!

Elisabeth Fink Words in non-periodic branch groups