Words in non-periodic branch groups · branch groups Elisabeth Fink Introduction Groups acting on Rooted Trees Rooted Trees Automorphisms A Construction Construction Words An Example

Words innon-periodic

branchgroups

ElisabethFink

Introduction

Groupsacting onRooted TreesRooted Trees

Automorphisms

AConstruction

ConstructionWordsAn Example

Lemmata and aTheorem

Growth

FurtherQuestions

Words in non-periodic branch groups

Elisabeth Fink

University of Oxford

May 28, 2013

Elisabeth Fink Words in non-periodic branch groups


branchgroups

ElisabethFink

Introduction


Automorphisms

AConstruction



Growth

FurtherQuestions

Introduction

A construction of a branch group G withno non-abelian free subgroupsexponential growth

For any g, h ∈ G we construct wg,h(x , y) ∈ F (x , y) withwg,h(g, h) = 1 ∈ G.



branchgroups

ElisabethFink

Introduction


Automorphisms

AConstruction



Growth

FurtherQuestions

Introduction

A construction of a branch group G with

no non-abelian free subgroupsexponential growth




branchgroups

ElisabethFink

Introduction


Automorphisms

AConstruction



Growth

FurtherQuestions

Introduction

A construction of a branch group G withno non-abelian free subgroups

exponential growthFor any g, h ∈ G we construct wg,h(x , y) ∈ F (x , y) withwg,h(g, h) = 1 ∈ G.



branchgroups

ElisabethFink

Introduction


Automorphisms

AConstruction



Growth

FurtherQuestions

Introduction





branchgroups

ElisabethFink

Introduction


Automorphisms

AConstruction



Growth

FurtherQuestions

Introduction





branchgroups

ElisabethFink

Introduction


Automorphisms

AConstruction



Growth

FurtherQuestions

Rooted Trees

Rooted tree: cyclefree graph with vertices V and edges Edistinguished root: rdistance function: d(v ,w), number of edges in unique path from vto wlevel n: Ω(n) = v ∈ V : d(v , r) = n

Tn = (V ′,E ′) with

V ′ = v ∈ V : d(v , r) ≤ nE ′ = e ∈ E : e = evw , v ,w ∈ V ′

Tv : subtree with root v ∈ V



branchgroups

ElisabethFink

Introduction


Automorphisms

AConstruction



Growth

FurtherQuestions

Rooted Trees

Rooted tree: cyclefree graph with vertices V and edges E

distinguished root: rdistance function: d(v ,w), number of edges in unique path from vto wlevel n: Ω(n) = v ∈ V : d(v , r) = n






branchgroups

ElisabethFink

Introduction


Automorphisms

AConstruction



Growth

FurtherQuestions

Rooted Trees

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

distance function: d(v ,w), number of edges in unique path from vto wlevel n: Ω(n) = v ∈ V : d(v , r) = n






branchgroups

ElisabethFink

Introduction


Automorphisms

AConstruction



Growth

FurtherQuestions

Rooted Trees

Rooted tree: cyclefree graph with vertices V and edges Edistinguished root: rdistance function: d(v ,w), number of edges in unique path from vto w

level n: Ω(n) = v ∈ V : d(v , r) = n






branchgroups

ElisabethFink

Introduction


Automorphisms

AConstruction



Growth

FurtherQuestions

Rooted Trees







branchgroups

ElisabethFink

Introduction


Automorphisms

AConstruction



Growth

FurtherQuestions

Rooted Trees



V ′ = v ∈ V : d(v , r) ≤ n

E ′ = e ∈ E : e = evw , v ,w ∈ V ′Tv : subtree with root v ∈ V



branchgroups

ElisabethFink

Introduction


Automorphisms

AConstruction



Growth

FurtherQuestions

Rooted Trees







branchgroups

ElisabethFink

Introduction


Automorphisms

AConstruction



Growth

FurtherQuestions

Rooted Trees







branchgroups

ElisabethFink

Introduction


Automorphisms

AConstruction



Growth

FurtherQuestions

Rooted Trees







branchgroups

ElisabethFink

Introduction


Automorphisms

AConstruction



Growth

FurtherQuestions

Automorphisms acting on rooted trees

acting on vertices, preserve edge incidence and root

stG(n) = g ∈ G : g acts trivially on TnrstG(v) = g ∈ G : g acts trivially on T\Tv

rstG(n) =Q

v∈Ω(n) rstG(v)

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



branchgroups

ElisabethFink

Introduction


Automorphisms

AConstruction



Growth

FurtherQuestions




rstG(n) =Q

v∈Ω(n) rstG(v)




branchgroups

ElisabethFink

Introduction


Automorphisms

AConstruction



Growth

FurtherQuestions



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)




branchgroups

ElisabethFink

Introduction


Automorphisms

AConstruction



Growth

FurtherQuestions




rstG(n) =Q

v∈Ω(n) rstG(v)




branchgroups

ElisabethFink

Introduction


Automorphisms

AConstruction



Growth

FurtherQuestions




rstG(n) =Q

v∈Ω(n) rstG(v)




branchgroups

ElisabethFink

Introduction


Automorphisms

AConstruction



Growth

FurtherQuestions




rstG(n) =Q

v∈Ω(n) rstG(v)




branchgroups

ElisabethFink

Introduction


Automorphisms

AConstruction



Growth

FurtherQuestions




rstG(n) =Q

v∈Ω(n) rstG(v)




branchgroups

ElisabethFink

Introduction


Automorphisms

AConstruction



Growth

FurtherQuestions




rstG(n) =Q

v∈Ω(n) rstG(v)




branchgroups

ElisabethFink

Introduction


Automorphisms

AConstruction



Growth

FurtherQuestions

A specific construction

Finite cyclic groups Aii∈N, 〈ai〉 = Ai

|Ai | = pi , (pi , pj ) = 1 for i 6= j

defining sequence pi

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



branchgroups

ElisabethFink

Introduction


Automorphisms

AConstruction



Growth

FurtherQuestions


Finite cyclic groups Aii∈N,

〈ai〉 = Ai

|Ai | = pi , (pi , pj ) = 1 for i 6= j





branchgroups

ElisabethFink

Introduction


Automorphisms

AConstruction



Growth

FurtherQuestions



|Ai | = pi , (pi , pj ) = 1 for i 6= j





branchgroups

ElisabethFink

Introduction


Automorphisms

AConstruction



Growth

FurtherQuestions



|Ai | = pi , (pi , pj ) = 1 for i 6= j





branchgroups

ElisabethFink

Introduction


Automorphisms

AConstruction



Growth

FurtherQuestions



|Ai | = pi , (pi , pj ) = 1 for i 6= j





branchgroups

ElisabethFink

Introduction


Automorphisms

AConstruction



Growth

FurtherQuestions



|Ai | = pi , (pi , pj ) = 1 for i 6= j





branchgroups

ElisabethFink

Introduction


Automorphisms

AConstruction



Growth

FurtherQuestions



|Ai | = pi , (pi , pj ) = 1 for i 6= j


G = 〈a, b〉 with

rooted automorphism a0,cyclically permutes Ω(1)

spinal automorphism brecursively defined asbn = (bn+1, an+1, 1, . . . , 1)1

abi

b1

b2

b3

a1

a2

a3b4 a4



branchgroups

ElisabethFink

Introduction


Automorphisms

AConstruction



Growth

FurtherQuestions



|Ai | = pi , (pi , pj ) = 1 for i 6= j


G = 〈a, b〉 with

rooted automorphism a0

,cyclically permutes Ω(1)


abi

b1

b2

b3

a1

a2

a3b4 a4



branchgroups

ElisabethFink

Introduction


Automorphisms

AConstruction



Growth

FurtherQuestions



|Ai | = pi , (pi , pj ) = 1 for i 6= j


G = 〈a, b〉 with



abi

b1

b2

b3

a1

a2

a3b4 a4



branchgroups

ElisabethFink

Introduction


Automorphisms

AConstruction



Growth

FurtherQuestions



|Ai | = pi , (pi , pj ) = 1 for i 6= j


G = 〈a, b〉 with


spinal automorphism b

recursively defined asbn = (bn+1, an+1, 1, . . . , 1)1

abi

b1

b2

b3

a1

a2

a3b4 a4



branchgroups

ElisabethFink

Introduction


Automorphisms

AConstruction



Growth

FurtherQuestions



|Ai | = pi , (pi , pj ) = 1 for i 6= j


G = 〈a, b〉 with



abi

b1

b2

b3

a1

a2

a3b4 a4



branchgroups

ElisabethFink

Introduction


Automorphisms

AConstruction



Growth

FurtherQuestions



|Ai | = pi , (pi , pj ) = 1 for i 6= j


G = 〈a, b〉 with



abi

b1

b2

b3

a1

a2

a3b4 a4



branchgroups

ElisabethFink

Introduction


Automorphisms

AConstruction



Growth

FurtherQuestions

Basic properties of G

Proposition

G acts on Ω(n) as An−1 o · · · o A0

Gab = Cp0 × C∞, indirectly implies G is not just infinite1 6= N C G, then N is finitely generatedN C G, N non-trivial, then G/N is solubleG has infinite virtual first betti numberG is not largeG(n+1) ≤ rstG(n)



branchgroups

ElisabethFink

Introduction


Automorphisms

AConstruction



Growth

FurtherQuestions


Proposition





branchgroups

ElisabethFink

Introduction


Automorphisms

AConstruction



Growth

FurtherQuestions


Proposition


Gab = Cp0 × C∞, indirectly implies G is not just infinite

1 6= N C G, then N is finitely generatedN C G, N non-trivial, then G/N is solubleG has infinite virtual first betti numberG is not largeG(n+1) ≤ rstG(n)



branchgroups

ElisabethFink

Introduction


Automorphisms

AConstruction



Growth

FurtherQuestions


Proposition


Gab = Cp0 × C∞, indirectly implies G is not just infinite1 6= N C G, then N is finitely generated

N C G, N non-trivial, then G/N is solubleG has infinite virtual first betti numberG is not largeG(n+1) ≤ rstG(n)



branchgroups

ElisabethFink

Introduction


Automorphisms

AConstruction



Growth

FurtherQuestions


Proposition


Gab = Cp0 × C∞, indirectly implies G is not just infinite1 6= N C G, then N is finitely generatedN C G, N non-trivial, then G/N is soluble

G has infinite virtual first betti numberG is not largeG(n+1) ≤ rstG(n)



branchgroups

ElisabethFink

Introduction


Automorphisms

AConstruction



Growth

FurtherQuestions


Proposition


Gab = Cp0 × C∞, indirectly implies G is not just infinite1 6= N C G, then N is finitely generatedN C G, N non-trivial, then G/N is solubleG has infinite virtual first betti number

G is not largeG(n+1) ≤ rstG(n)



branchgroups

ElisabethFink

Introduction


Automorphisms

AConstruction



Growth

FurtherQuestions


Proposition


Gab = Cp0 × C∞, indirectly implies G is not just infinite1 6= N C G, then N is finitely generatedN C G, N non-trivial, then G/N is solubleG has infinite virtual first betti numberG is not large

G(n+1) ≤ rstG(n)



branchgroups

ElisabethFink

Introduction


Automorphisms

AConstruction



Growth

FurtherQuestions


Proposition





branchgroups

ElisabethFink

Introduction


Automorphisms

AConstruction



Growth

FurtherQuestions

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−11 , a1, 1, . . . , 1

´1,

b−11 b1a−1

1 a1 1



branchgroups

ElisabethFink

Introduction


Automorphisms

AConstruction



Growth

FurtherQuestions

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−11 , a1, 1, . . . , 1

´1,

b−11 b1a−1

1 a1 1



branchgroups

ElisabethFink

Introduction


Automorphisms

AConstruction



Growth

FurtherQuestions

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−11 , a1, 1, . . . , 1

´1,

b−11 b1a−1

1 a1 1



branchgroups

ElisabethFink

Introduction


Automorphisms

AConstruction



Growth

FurtherQuestions

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−11 , a1, 1, . . . , 1

´1,

b−11 b1a−1

1 a1 1



branchgroups

ElisabethFink

Introduction


Automorphisms

AConstruction



Growth

FurtherQuestions

An Example

g1 = (ab)4a−4, g2 = [b, ba]

b1 b1a1 b1a1b1a1 a1 1 b−11 b1a−1

1a1 1

c1 = [g1, g2] =“

1,“

a2, b−12 , a−1

2 , 1, . . . , b2

”,“

1, b−12 , a−1

2 b2, a2, 1, . . . , 1”

, 1, . . . , 1”

2

a2 b−12 a−1

2 1 b2 1 b2 a−12 b2 a2 1



branchgroups

ElisabethFink

Introduction


Automorphisms

AConstruction



Growth

FurtherQuestions

An Example

g1 = (ab)4a−4, g2 = [b, ba]

b1 b1a1 b1a1b1a1 a1 1 b−11 b1a−1

1a1 1

c1 = [g1, g2] =“

1,“

a2, b−12 , a−1

2 , 1, . . . , b2

”,“

1, b−12 , a−1

2 b2, a2, 1, . . . , 1”

, 1, . . . , 1”

2

a2 b−12 a−1

2 1 b2 1 b2 a−12 b2 a2 1



branchgroups

ElisabethFink

Introduction


Automorphisms

AConstruction



Growth

FurtherQuestions

An Example

g1 = (ab)4a−4, g2 = [b, ba]

b1 b1a1 b1a1b1a1 a1 1 b−11 b1a−1

1a1 1

c1 = [g1, g2] =“

1,“

a2, b−12 , a−1

2 , 1, . . . , b2

”,“

1, b−12 , a−1

2 b2, a2, 1, . . . , 1”

, 1, . . . , 1”

2

a2 b−12 a−1

2 1 b2 1 b2 a−12 b2 a2 1



branchgroups

ElisabethFink

Introduction


Automorphisms

AConstruction



Growth

FurtherQuestions

An Example

c1 = [g1, g2]

a2 b−12 a−1

2 1 b2 1 b2a−12 b2a2 1

cg41

1 = [g1, g2]g41 =

“1,“

1, 1, 1, b2, a2, b−12 , a−1

2 b−12 a−1

2 b2a2, 1, . . . , 1”

,“1, 1, 1, 1, b−1

2 , a−12 b−1

2 a−12 b2

2a2, a−12 b−1

2 a2b2a2, 1, . . . , 1”

, 1, . . . , 1”

2.

1 1 1 b2 a2 a−12 b−1

2 a−12 b2a2 1 1 1 1 b−1

21 a−12 b−1

2 a−12 b2

2a2 a−12 b−1

2 a2b2a2 1



branchgroups

ElisabethFink

Introduction


Automorphisms

AConstruction



Growth

FurtherQuestions

An Example

c1 = [g1, g2]

a2 b−12 a−1

2 1 b2 1 b2a−12 b2a2 1

cg41

1 = [g1, g2]g41

=“

1,“

1, 1, 1, b2, a2, b−12 , a−1

2 b−12 a−1

2 b2a2, 1, . . . , 1”

,“1, 1, 1, 1, b−1

2 , a−12 b−1

2 a−12 b2

2a2, a−12 b−1

2 a2b2a2, 1, . . . , 1”

, 1, . . . , 1”

2.

1 1 1 b2 a2 a−12 b−1

2 a−12 b2a2 1 1 1 1 b−1

21 a−12 b−1

2 a−12 b2

2a2 a−12 b−1

2 a2b2a2 1



branchgroups

ElisabethFink

Introduction


Automorphisms

AConstruction



Growth

FurtherQuestions

An Example

c1 = [g1, g2]

a2 b−12 a−1

2 1 b2 1 b2a−12 b2a2 1

cg41

1 = [g1, g2]g41 =

“1,“

1, 1, 1, b2, a2, b−12 , a−1

2 b−12 a−1

2 b2a2, 1, . . . , 1”

,“1, 1, 1, 1, b−1

2 , a−12 b−1

2 a−12 b2

2a2, a−12 b−1

2 a2b2a2, 1, . . . , 1”

, 1, . . . , 1”

2.

1 1 1 b2 a2 a−12 b−1

2 a−12 b2a2 1 1 1 1 b−1

21 a−12 b−1

2 a−12 b2

2a2 a−12 b−1

2 a2b2a2 1



branchgroups

ElisabethFink

Introduction


Automorphisms

AConstruction



Growth

FurtherQuestions

An Example

c1 = [g1, g2]

a2 b−12 a−1

2 1 b2 1 b2a−12 b2a2 1

cg41

1 = [g1, g2]g41 =

“1,“

1, 1, 1, b2, a2, b−12 , a−1

2 b−12 a−1

2 b2a2, 1, . . . , 1”

,“1, 1, 1, 1, b−1

2 , a−12 b−1

2 a−12 b2

2a2, a−12 b−1

2 a2b2a2, 1, . . . , 1”

, 1, . . . , 1”

2.

1 1 1 b2 a2 a−12 b−1

2 a−12 b2a2 1 1 1 1 b−1

21 a−12 b−1

2 a−12 b2

2a2 a−12 b−1

2 a2b2a2 1



branchgroups

ElisabethFink

Introduction


Automorphisms

AConstruction



Growth

FurtherQuestions

An Example

c1 = [g1, g2]

a2 b−12 a−1

2 1 b2 1 b2a−12 b2a2 1

cg41

1 = [g1, g2]g41

1 1 1 b2 a2 a−12 b−1

2 a−12 b2a2 1 1 1 1 b−1

21 a−12 b−1

2 a−12 b2

2a2 a−12 b−1

2 a2b2a2 1

Comparing pictures:[c1, cg4

11

]= 1

wg1,g2 (x , y) =[[x , y ], [x , y ]x

4]

with wg1,g2 (g1, g2) = 1



branchgroups

ElisabethFink

Introduction


Automorphisms

AConstruction



Growth

FurtherQuestions

An Example

c1 = [g1, g2]

a2 b−12 a−1

2 1 b2 1 b2a−12 b2a2 1

cg41

1 = [g1, g2]g41

1 1 1 b2 a2 a−12 b−1

2 a−12 b2a2 1 1 1 1 b−1

21 a−12 b−1

2 a−12 b2

2a2 a−12 b−1

2 a2b2a2 1


11

]= 1

wg1,g2 (x , y) =[[x , y ], [x , y ]x

4]

with wg1,g2 (g1, g2) = 1



branchgroups

ElisabethFink

Introduction


Automorphisms

AConstruction



Growth

FurtherQuestions

An Example

c1 = [g1, g2]

a2 b−12 a−1

2 1 b2 1 b2a−12 b2a2 1

cg41

1 = [g1, g2]g41

1 1 1 b2 a2 a−12 b−1

2 a−12 b2a2 1 1 1 1 b−1

21 a−12 b−1

2 a−12 b2

2a2 a−12 b−1

2 a2b2a2 1


11

]= 1

wg1,g2 (x , y) =[[x , y ], [x , y ]x

4]

with wg1,g2 (g1, g2) = 1



branchgroups

ElisabethFink

Introduction


Automorphisms

AConstruction



Growth

FurtherQuestions

Lemmata

DefinitionLet 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

nsg : g = ak ·

Qsgi=1 a−ki bqi aki

o

c0 = g, c1 = [g1, g2] , ci =hci−1, c

ci−2i−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 !



branchgroups

ElisabethFink

Introduction


Automorphisms

AConstruction



Growth

FurtherQuestions

Lemmata

DefinitionLet 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

nsg : g = ak ·


o

c0 = g, c1 = [g1, g2] , ci =hci−1, c

ci−2i−1

i, i ≥ 2

Lemma

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

ci ∈ rstG(i − 1) ≤ stG(i − 1), i ≥ 2




branchgroups

ElisabethFink

Introduction


Automorphisms

AConstruction



Growth

FurtherQuestions

Lemmata


nsg : g = ak ·


o

c0 = g, c1 = [g1, g2] , ci =hci−1, c

ci−2i−1

i, i ≥ 2

Lemma

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

ci ∈ rstG(i − 1) ≤ stG(i − 1), i ≥ 2




branchgroups

ElisabethFink

Introduction


Automorphisms

AConstruction



Growth

FurtherQuestions

Lemmata


nsg : g = ak ·


o

c0 = g, c1 = [g1, g2] ,

ci =hci−1, c

ci−2i−1

i, i ≥ 2

Lemma

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

ci ∈ rstG(i − 1) ≤ stG(i − 1), i ≥ 2




branchgroups

ElisabethFink

Introduction


Automorphisms

AConstruction



Growth

FurtherQuestions

Lemmata


nsg : g = ak ·


o

c0 = g, c1 = [g1, g2] , ci =hci−1, c

ci−2i−1

i, i ≥ 2

Lemma

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

ci ∈ rstG(i − 1) ≤ stG(i − 1), i ≥ 2




branchgroups

ElisabethFink

Introduction


Automorphisms

AConstruction



Growth

FurtherQuestions

Lemmata


nsg : g = ak ·


o

c0 = g, c1 = [g1, g2] , ci =hci−1, c

ci−2i−1

i, i ≥ 2

Lemma

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

ci ∈ rstG(i − 1) ≤ stG(i − 1), i ≥ 2




branchgroups

ElisabethFink

Introduction


Automorphisms

AConstruction



Growth

FurtherQuestions

Lemmata


nsg : g = ak ·


o

c0 = g, c1 = [g1, g2] , ci =hci−1, c

ci−2i−1

i, i ≥ 2

Lemma

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

ci ∈ rstG(i − 1) ≤ stG(i − 1), i ≥ 2




branchgroups

ElisabethFink

Introduction


Automorphisms

AConstruction



Growth

FurtherQuestions

Lemmata


nsg : g = ak ·


o

c0 = g, c1 = [g1, g2] , ci =hci−1, c

ci−2i−1

i, i ≥ 2

Lemma

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

ci ∈ rstG(i − 1) ≤ stG(i − 1), i ≥ 2




branchgroups

ElisabethFink

Introduction


Automorphisms

AConstruction



Growth

FurtherQuestions

Theorem

c0 = g, c1 = [g1, g2] , ci =hci−1, c

ci−2i−1

i, i ≥ 2

Proposition

ci has the form ci =`di,1, . . . , di,mi

í with di,j one of the following

1 di,j = bti , t ∈ Z

2 di,j = aqb, q 6= 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)3Qi−1

k=0 pk , then for each g, h ∈ G there exists a wordwg,h(x , y) ∈ F (x , y) with wg,h(g, h) = 1 ∈ G. Hence G has nonon-abelian free subgroups.



branchgroups

ElisabethFink

Introduction


Automorphisms

AConstruction



Growth

FurtherQuestions

Theorem

c0 = g, c1 = [g1, g2] , ci =hci−1, c

ci−2i−1

i, i ≥ 2

Proposition






4 di,j = 1


Theorem

If pi ≥ (25pi−1)3Qi−1




branchgroups

ElisabethFink

Introduction


Automorphisms

AConstruction



Growth

FurtherQuestions

Theorem

c0 = g, c1 = [g1, g2] , ci =hci−1, c

ci−2i−1

i, i ≥ 2

Proposition






4 di,j = 1


Theorem

If pi ≥ (25pi−1)3Qi−1




branchgroups

ElisabethFink

Introduction


Automorphisms

AConstruction



Growth

FurtherQuestions

Theorem

c0 = g, c1 = [g1, g2] , ci =hci−1, c

ci−2i−1

i, i ≥ 2

Proposition






4 di,j = 1


Theorem

If pi ≥ (25pi−1)3Qi−1




branchgroups

ElisabethFink

Introduction


Automorphisms

AConstruction



Growth

FurtherQuestions

Theorem

c0 = g, c1 = [g1, g2] , ci =hci−1, c

ci−2i−1

i, i ≥ 2

Proposition






4 di,j = 1


Theorem

If pi ≥ (25pi−1)3Qi−1




branchgroups

ElisabethFink

Introduction


Automorphisms

AConstruction



Growth

FurtherQuestions

Theorem

c0 = g, c1 = [g1, g2] , ci =hci−1, c

ci−2i−1

i, i ≥ 2

Proposition






4 di,j = 1


Theorem

If pi ≥ (25pi−1)3Qi−1




branchgroups

ElisabethFink

Introduction


Automorphisms

AConstruction



Growth

FurtherQuestions

Theorem

c0 = g, c1 = [g1, g2] , ci =hci−1, c

ci−2i−1

i, i ≥ 2

Proposition






4 di,j = 1


Theorem

If pi ≥ (25pi−1)3Qi−1

k=0 pk , then for each g, h ∈ G there exists a wordwg,h(x , y) ∈ F (x , y) with wg,h(g, h) = 1 ∈ G.

Hence G has nonon-abelian free subgroups.



branchgroups

ElisabethFink

Introduction


Automorphisms

AConstruction



Growth

FurtherQuestions

Theorem

c0 = g, c1 = [g1, g2] , ci =hci−1, c

ci−2i−1

i, i ≥ 2

Proposition






4 di,j = 1


Theorem

If pi ≥ (25pi−1)3Qi−1




branchgroups

ElisabethFink

Introduction


Automorphisms

AConstruction



Growth

FurtherQuestions

Growth

Proposition

G does not have polynomial growth.

Self-SimilaritiesFind copies of G1 in G:

G1 = 〈a1, b1〉 G1 = 〈a1, b1〉

b1a1 a1 b1b1 b1a1



branchgroups

ElisabethFink

Introduction


Automorphisms

AConstruction



Growth

FurtherQuestions

Growth

Proposition



G1 = 〈a1, b1〉 G1 = 〈a1, b1〉

b1a1 a1 b1b1 b1a1



branchgroups

ElisabethFink

Introduction


Automorphisms

AConstruction



Growth

FurtherQuestions

Growth

Proposition


Self-Similarities

Find copies of G1 in G:

G1 = 〈a1, b1〉 G1 = 〈a1, b1〉

b1a1 a1 b1b1 b1a1



branchgroups

ElisabethFink

Introduction


Automorphisms

AConstruction



Growth

FurtherQuestions

Growth

Proposition



G1 = 〈a1, b1〉 G1 = 〈a1, b1〉

b1a1 a1 b1b1 b1a1



branchgroups

ElisabethFink

Introduction


Automorphisms

AConstruction



Growth

FurtherQuestions

Growth

Proposition



G1 = 〈a1, b1〉 G1 = 〈a1, b1〉

b1a1 a1 b1b1 b1a1



branchgroups

ElisabethFink

Introduction


Automorphisms

AConstruction



Growth

FurtherQuestions

Growth

Lemma

γG

“Qik=1 p2

k n”≥ γGi (n)

Qik=1 pk /3.

Proposition

γBi (2pi ) ≥ 2pi−1 · (pi − 1)pi2 −2.

Theorem

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

5

í ·Qi

k=0 pk for all sufficiently large i ,then G has exponential growth.



branchgroups

ElisabethFink

Introduction


Automorphisms

AConstruction



Growth

FurtherQuestions

Growth

Lemma

γG

“Qik=1 p2

k n”≥ γGi (n)

Qik=1 pk /3.

Proposition

γBi (2pi ) ≥ 2pi−1 · (pi − 1)pi2 −2.

Theorem


5

í ·Qi




branchgroups

ElisabethFink

Introduction


Automorphisms

AConstruction



Growth

FurtherQuestions

Growth

Lemma

γG

“Qik=1 p2

k n”≥ γGi (n)

Qik=1 pk /3.

Proposition

γBi (2pi ) ≥ 2pi−1 · (pi − 1)pi2 −2.

Theorem


5

í ·Qi




branchgroups

ElisabethFink

Introduction


Automorphisms

AConstruction



Growth

FurtherQuestions

Growth

Lemma

γG

“Qik=1 p2

k n”≥ γGi (n)

Qik=1 pk /3.

Proposition

γBi (2pi ) ≥ 2pi−1 · (pi − 1)pi2 −2.

Theorem


5

í ·Qi




branchgroups

ElisabethFink

Introduction


Automorphisms

AConstruction



Growth

FurtherQuestions

Further Questions

Is G amenable?

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



branchgroups

ElisabethFink

Introduction


Automorphisms

AConstruction



Growth

FurtherQuestions

Further Questions

Is G amenable?Growth in all cases?

Similar examples are amenable for slow growth(Brieussel)Does G contain a free semigroup?



branchgroups

ElisabethFink

Introduction


Automorphisms

AConstruction



Growth

FurtherQuestions

Further Questions

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

Does G contain a free semigroup?



branchgroups

ElisabethFink

Introduction


Automorphisms

AConstruction



Growth

FurtherQuestions

Further Questions

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



branchgroups

ElisabethFink

Introduction


Automorphisms

AConstruction



Growth

FurtherQuestions

Thank you!


Words in non-periodic branch groups · branch groups Elisabeth Fink Introduction Groups acting on Rooted Trees Rooted Trees Automorphisms A Construction Construction Words An Example

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