Words in non-periodic branch groups · branch groups Elisabeth Fink Introduction Groups acting on...
Transcript of Words in non-periodic branch groups · branch groups Elisabeth Fink Introduction Groups acting on...
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
Words innon-periodic
branchgroups
ElisabethFink
Introduction
Groupsacting onRooted TreesRooted Trees
Automorphisms
AConstruction
ConstructionWordsAn Example
Lemmata and aTheorem
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.
Elisabeth Fink Words in non-periodic branch groups
Words innon-periodic
branchgroups
ElisabethFink
Introduction
Groupsacting onRooted TreesRooted Trees
Automorphisms
AConstruction
ConstructionWordsAn Example
Lemmata and aTheorem
Growth
FurtherQuestions
Introduction
A construction of a branch group G with
no 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.
Elisabeth Fink Words in non-periodic branch groups
Words innon-periodic
branchgroups
ElisabethFink
Introduction
Groupsacting onRooted TreesRooted Trees
Automorphisms
AConstruction
ConstructionWordsAn Example
Lemmata and aTheorem
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.
Elisabeth Fink Words in non-periodic branch groups
Words innon-periodic
branchgroups
ElisabethFink
Introduction
Groupsacting onRooted TreesRooted Trees
Automorphisms
AConstruction
ConstructionWordsAn Example
Lemmata and aTheorem
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.
Elisabeth Fink Words in non-periodic branch groups
Words innon-periodic
branchgroups
ElisabethFink
Introduction
Groupsacting onRooted TreesRooted Trees
Automorphisms
AConstruction
ConstructionWordsAn Example
Lemmata and aTheorem
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.
Elisabeth Fink Words in non-periodic branch groups
Words innon-periodic
branchgroups
ElisabethFink
Introduction
Groupsacting onRooted TreesRooted Trees
Automorphisms
AConstruction
ConstructionWordsAn Example
Lemmata and aTheorem
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
Elisabeth Fink Words in non-periodic branch groups
Words innon-periodic
branchgroups
ElisabethFink
Introduction
Groupsacting onRooted TreesRooted Trees
Automorphisms
AConstruction
ConstructionWordsAn Example
Lemmata and aTheorem
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
Tn = (V ′,E ′) with
V ′ = v ∈ V : d(v , r) ≤ nE ′ = e ∈ E : e = evw , v ,w ∈ V ′
Tv : subtree with root v ∈ V
Elisabeth Fink Words in non-periodic branch groups
Words innon-periodic
branchgroups
ElisabethFink
Introduction
Groupsacting onRooted TreesRooted Trees
Automorphisms
AConstruction
ConstructionWordsAn Example
Lemmata and aTheorem
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
Tn = (V ′,E ′) with
V ′ = v ∈ V : d(v , r) ≤ nE ′ = e ∈ E : e = evw , v ,w ∈ V ′
Tv : subtree with root v ∈ V
Elisabeth Fink Words in non-periodic branch groups
Words innon-periodic
branchgroups
ElisabethFink
Introduction
Groupsacting onRooted TreesRooted Trees
Automorphisms
AConstruction
ConstructionWordsAn Example
Lemmata and aTheorem
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
Tn = (V ′,E ′) with
V ′ = v ∈ V : d(v , r) ≤ nE ′ = e ∈ E : e = evw , v ,w ∈ V ′
Tv : subtree with root v ∈ V
Elisabeth Fink Words in non-periodic branch groups
Words innon-periodic
branchgroups
ElisabethFink
Introduction
Groupsacting onRooted TreesRooted Trees
Automorphisms
AConstruction
ConstructionWordsAn Example
Lemmata and aTheorem
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
Elisabeth Fink Words in non-periodic branch groups
Words innon-periodic
branchgroups
ElisabethFink
Introduction
Groupsacting onRooted TreesRooted Trees
Automorphisms
AConstruction
ConstructionWordsAn Example
Lemmata and aTheorem
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) ≤ n
E ′ = e ∈ E : e = evw , v ,w ∈ V ′Tv : subtree with root v ∈ V
Elisabeth Fink Words in non-periodic branch groups
Words innon-periodic
branchgroups
ElisabethFink
Introduction
Groupsacting onRooted TreesRooted Trees
Automorphisms
AConstruction
ConstructionWordsAn Example
Lemmata and aTheorem
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
Elisabeth Fink Words in non-periodic branch groups
Words innon-periodic
branchgroups
ElisabethFink
Introduction
Groupsacting onRooted TreesRooted Trees
Automorphisms
AConstruction
ConstructionWordsAn Example
Lemmata and aTheorem
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
Elisabeth Fink Words in non-periodic branch groups
Words innon-periodic
branchgroups
ElisabethFink
Introduction
Groupsacting onRooted TreesRooted Trees
Automorphisms
AConstruction
ConstructionWordsAn Example
Lemmata and aTheorem
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
Elisabeth Fink Words in non-periodic branch groups
Words innon-periodic
branchgroups
ElisabethFink
Introduction
Groupsacting onRooted TreesRooted Trees
Automorphisms
AConstruction
ConstructionWordsAn Example
Lemmata and aTheorem
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.
Elisabeth Fink Words in non-periodic branch groups
Words innon-periodic
branchgroups
ElisabethFink
Introduction
Groupsacting onRooted TreesRooted Trees
Automorphisms
AConstruction
ConstructionWordsAn Example
Lemmata and aTheorem
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.
Elisabeth Fink Words in non-periodic branch groups
Words innon-periodic
branchgroups
ElisabethFink
Introduction
Groupsacting onRooted TreesRooted Trees
Automorphisms
AConstruction
ConstructionWordsAn Example
Lemmata and aTheorem
Growth
FurtherQuestions
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)
DefinitionA group G is a branch group if it acts transitively on each level of a treeand each rstG(n) has finite index.
Elisabeth Fink Words in non-periodic branch groups
Words innon-periodic
branchgroups
ElisabethFink
Introduction
Groupsacting onRooted TreesRooted Trees
Automorphisms
AConstruction
ConstructionWordsAn Example
Lemmata and aTheorem
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.
Elisabeth Fink Words in non-periodic branch groups
Words innon-periodic
branchgroups
ElisabethFink
Introduction
Groupsacting onRooted TreesRooted Trees
Automorphisms
AConstruction
ConstructionWordsAn Example
Lemmata and aTheorem
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.
Elisabeth Fink Words in non-periodic branch groups
Words innon-periodic
branchgroups
ElisabethFink
Introduction
Groupsacting onRooted TreesRooted Trees
Automorphisms
AConstruction
ConstructionWordsAn Example
Lemmata and aTheorem
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.
Elisabeth Fink Words in non-periodic branch groups
Words innon-periodic
branchgroups
ElisabethFink
Introduction
Groupsacting onRooted TreesRooted Trees
Automorphisms
AConstruction
ConstructionWordsAn Example
Lemmata and aTheorem
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.
Elisabeth Fink Words in non-periodic branch groups
Words innon-periodic
branchgroups
ElisabethFink
Introduction
Groupsacting onRooted TreesRooted Trees
Automorphisms
AConstruction
ConstructionWordsAn Example
Lemmata and aTheorem
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.
Elisabeth Fink Words in non-periodic branch groups
Words innon-periodic
branchgroups
ElisabethFink
Introduction
Groupsacting onRooted TreesRooted Trees
Automorphisms
AConstruction
ConstructionWordsAn Example
Lemmata and aTheorem
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.
Elisabeth Fink Words in non-periodic branch groups
Words innon-periodic
branchgroups
ElisabethFink
Introduction
Groupsacting onRooted TreesRooted Trees
Automorphisms
AConstruction
ConstructionWordsAn Example
Lemmata and aTheorem
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.
Elisabeth Fink Words in non-periodic branch groups
Words innon-periodic
branchgroups
ElisabethFink
Introduction
Groupsacting onRooted TreesRooted Trees
Automorphisms
AConstruction
ConstructionWordsAn Example
Lemmata and aTheorem
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.
Elisabeth Fink Words in non-periodic branch groups
Words innon-periodic
branchgroups
ElisabethFink
Introduction
Groupsacting onRooted TreesRooted Trees
Automorphisms
AConstruction
ConstructionWordsAn Example
Lemmata and aTheorem
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.
Elisabeth Fink Words in non-periodic branch groups
Words innon-periodic
branchgroups
ElisabethFink
Introduction
Groupsacting onRooted TreesRooted Trees
Automorphisms
AConstruction
ConstructionWordsAn Example
Lemmata and aTheorem
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.
Elisabeth Fink Words in non-periodic branch groups
Words innon-periodic
branchgroups
ElisabethFink
Introduction
Groupsacting onRooted TreesRooted Trees
Automorphisms
AConstruction
ConstructionWordsAn Example
Lemmata and aTheorem
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.
Elisabeth Fink Words in non-periodic branch groups
Words innon-periodic
branchgroups
ElisabethFink
Introduction
Groupsacting onRooted TreesRooted Trees
Automorphisms
AConstruction
ConstructionWordsAn Example
Lemmata and aTheorem
Growth
FurtherQuestions
A specific construction
Finite cyclic groups Aii∈N, 〈ai〉 = Ai
|Ai | = pi , (pi , pj ) = 1 for i 6= j
defining sequence pi
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
Elisabeth Fink Words in non-periodic branch groups
Words innon-periodic
branchgroups
ElisabethFink
Introduction
Groupsacting onRooted TreesRooted Trees
Automorphisms
AConstruction
ConstructionWordsAn Example
Lemmata and aTheorem
Growth
FurtherQuestions
A specific construction
Finite cyclic groups Aii∈N, 〈ai〉 = Ai
|Ai | = pi , (pi , pj ) = 1 for i 6= j
defining sequence pi
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
Elisabeth Fink Words in non-periodic branch groups
Words innon-periodic
branchgroups
ElisabethFink
Introduction
Groupsacting onRooted TreesRooted Trees
Automorphisms
AConstruction
ConstructionWordsAn Example
Lemmata and aTheorem
Growth
FurtherQuestions
A specific construction
Finite cyclic groups Aii∈N, 〈ai〉 = Ai
|Ai | = pi , (pi , pj ) = 1 for i 6= j
defining sequence pi
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
Elisabeth Fink Words in non-periodic branch groups
Words innon-periodic
branchgroups
ElisabethFink
Introduction
Groupsacting onRooted TreesRooted Trees
Automorphisms
AConstruction
ConstructionWordsAn Example
Lemmata and aTheorem
Growth
FurtherQuestions
A specific construction
Finite cyclic groups Aii∈N, 〈ai〉 = Ai
|Ai | = pi , (pi , pj ) = 1 for i 6= j
defining sequence pi
G = 〈a, b〉 with
rooted automorphism a0,cyclically permutes Ω(1)
spinal automorphism b
recursively defined asbn = (bn+1, an+1, 1, . . . , 1)1
abi
b1
b2
b3
a1
a2
a3b4 a4
Elisabeth Fink Words in non-periodic branch groups
Words innon-periodic
branchgroups
ElisabethFink
Introduction
Groupsacting onRooted TreesRooted Trees
Automorphisms
AConstruction
ConstructionWordsAn Example
Lemmata and aTheorem
Growth
FurtherQuestions
A specific construction
Finite cyclic groups Aii∈N, 〈ai〉 = Ai
|Ai | = pi , (pi , pj ) = 1 for i 6= j
defining sequence pi
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
Elisabeth Fink Words in non-periodic branch groups
Words innon-periodic
branchgroups
ElisabethFink
Introduction
Groupsacting onRooted TreesRooted Trees
Automorphisms
AConstruction
ConstructionWordsAn Example
Lemmata and aTheorem
Growth
FurtherQuestions
A specific construction
Finite cyclic groups Aii∈N, 〈ai〉 = Ai
|Ai | = pi , (pi , pj ) = 1 for i 6= j
defining sequence pi
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
Elisabeth Fink Words in non-periodic branch groups
Words innon-periodic
branchgroups
ElisabethFink
Introduction
Groupsacting onRooted TreesRooted Trees
Automorphisms
AConstruction
ConstructionWordsAn Example
Lemmata and aTheorem
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)
Elisabeth Fink Words in non-periodic branch groups
Words innon-periodic
branchgroups
ElisabethFink
Introduction
Groupsacting onRooted TreesRooted Trees
Automorphisms
AConstruction
ConstructionWordsAn Example
Lemmata and aTheorem
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)
Elisabeth Fink Words in non-periodic branch groups
Words innon-periodic
branchgroups
ElisabethFink
Introduction
Groupsacting onRooted TreesRooted Trees
Automorphisms
AConstruction
ConstructionWordsAn Example
Lemmata and aTheorem
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 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)
Elisabeth Fink Words in non-periodic branch groups
Words innon-periodic
branchgroups
ElisabethFink
Introduction
Groupsacting onRooted TreesRooted Trees
Automorphisms
AConstruction
ConstructionWordsAn Example
Lemmata and aTheorem
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 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)
Elisabeth Fink Words in non-periodic branch groups
Words innon-periodic
branchgroups
ElisabethFink
Introduction
Groupsacting onRooted TreesRooted Trees
Automorphisms
AConstruction
ConstructionWordsAn Example
Lemmata and aTheorem
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 soluble
G has infinite virtual first betti numberG is not largeG(n+1) ≤ rstG(n)
Elisabeth Fink Words in non-periodic branch groups
Words innon-periodic
branchgroups
ElisabethFink
Introduction
Groupsacting onRooted TreesRooted Trees
Automorphisms
AConstruction
ConstructionWordsAn Example
Lemmata and aTheorem
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 number
G is not largeG(n+1) ≤ rstG(n)
Elisabeth Fink Words in non-periodic branch groups
Words innon-periodic
branchgroups
ElisabethFink
Introduction
Groupsacting onRooted TreesRooted Trees
Automorphisms
AConstruction
ConstructionWordsAn Example
Lemmata and aTheorem
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 large
G(n+1) ≤ rstG(n)
Elisabeth Fink Words in non-periodic branch groups
Words innon-periodic
branchgroups
ElisabethFink
Introduction
Groupsacting onRooted TreesRooted Trees
Automorphisms
AConstruction
ConstructionWordsAn Example
Lemmata and aTheorem
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)
Elisabeth Fink Words in non-periodic branch groups
Words innon-periodic
branchgroups
ElisabethFink
Introduction
Groupsacting onRooted TreesRooted Trees
Automorphisms
AConstruction
ConstructionWordsAn Example
Lemmata and aTheorem
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
Elisabeth Fink Words in non-periodic branch groups
Words innon-periodic
branchgroups
ElisabethFink
Introduction
Groupsacting onRooted TreesRooted Trees
Automorphisms
AConstruction
ConstructionWordsAn Example
Lemmata and aTheorem
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
Elisabeth Fink Words in non-periodic branch groups
Words innon-periodic
branchgroups
ElisabethFink
Introduction
Groupsacting onRooted TreesRooted Trees
Automorphisms
AConstruction
ConstructionWordsAn Example
Lemmata and aTheorem
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
Elisabeth Fink Words in non-periodic branch groups
Words innon-periodic
branchgroups
ElisabethFink
Introduction
Groupsacting onRooted TreesRooted Trees
Automorphisms
AConstruction
ConstructionWordsAn Example
Lemmata and aTheorem
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
Elisabeth Fink Words in non-periodic branch groups
Words innon-periodic
branchgroups
ElisabethFink
Introduction
Groupsacting onRooted TreesRooted Trees
Automorphisms
AConstruction
ConstructionWordsAn Example
Lemmata and aTheorem
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
Elisabeth Fink Words in non-periodic branch groups
Words innon-periodic
branchgroups
ElisabethFink
Introduction
Groupsacting onRooted TreesRooted Trees
Automorphisms
AConstruction
ConstructionWordsAn Example
Lemmata and aTheorem
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
Elisabeth Fink Words in non-periodic branch groups
Words innon-periodic
branchgroups
ElisabethFink
Introduction
Groupsacting onRooted TreesRooted Trees
Automorphisms
AConstruction
ConstructionWordsAn Example
Lemmata and aTheorem
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
Elisabeth Fink Words in non-periodic branch groups
Words innon-periodic
branchgroups
ElisabethFink
Introduction
Groupsacting onRooted TreesRooted Trees
Automorphisms
AConstruction
ConstructionWordsAn Example
Lemmata and aTheorem
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
Elisabeth Fink Words in non-periodic branch groups
Words innon-periodic
branchgroups
ElisabethFink
Introduction
Groupsacting onRooted TreesRooted Trees
Automorphisms
AConstruction
ConstructionWordsAn Example
Lemmata and aTheorem
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
Elisabeth Fink Words in non-periodic branch groups
Words innon-periodic
branchgroups
ElisabethFink
Introduction
Groupsacting onRooted TreesRooted Trees
Automorphisms
AConstruction
ConstructionWordsAn Example
Lemmata and aTheorem
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
Elisabeth Fink Words in non-periodic branch groups
Words innon-periodic
branchgroups
ElisabethFink
Introduction
Groupsacting onRooted TreesRooted Trees
Automorphisms
AConstruction
ConstructionWordsAn Example
Lemmata and aTheorem
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
Elisabeth Fink Words in non-periodic branch groups
Words innon-periodic
branchgroups
ElisabethFink
Introduction
Groupsacting onRooted TreesRooted Trees
Automorphisms
AConstruction
ConstructionWordsAn Example
Lemmata and aTheorem
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
Elisabeth Fink Words in non-periodic branch groups
Words innon-periodic
branchgroups
ElisabethFink
Introduction
Groupsacting onRooted TreesRooted Trees
Automorphisms
AConstruction
ConstructionWordsAn Example
Lemmata and aTheorem
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
Elisabeth Fink Words in non-periodic branch groups
Words innon-periodic
branchgroups
ElisabethFink
Introduction
Groupsacting onRooted TreesRooted Trees
Automorphisms
AConstruction
ConstructionWordsAn Example
Lemmata and aTheorem
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
Elisabeth Fink Words in non-periodic branch groups
Words innon-periodic
branchgroups
ElisabethFink
Introduction
Groupsacting onRooted TreesRooted Trees
Automorphisms
AConstruction
ConstructionWordsAn Example
Lemmata and aTheorem
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 !
Elisabeth Fink Words in non-periodic branch groups
Words innon-periodic
branchgroups
ElisabethFink
Introduction
Groupsacting onRooted TreesRooted Trees
Automorphisms
AConstruction
ConstructionWordsAn Example
Lemmata and aTheorem
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 !
Elisabeth Fink Words in non-periodic branch groups
Words innon-periodic
branchgroups
ElisabethFink
Introduction
Groupsacting onRooted TreesRooted Trees
Automorphisms
AConstruction
ConstructionWordsAn Example
Lemmata and aTheorem
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 !
Elisabeth Fink Words in non-periodic branch groups
Words innon-periodic
branchgroups
ElisabethFink
Introduction
Groupsacting onRooted TreesRooted Trees
Automorphisms
AConstruction
ConstructionWordsAn Example
Lemmata and aTheorem
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 !
Elisabeth Fink Words in non-periodic branch groups
Words innon-periodic
branchgroups
ElisabethFink
Introduction
Groupsacting onRooted TreesRooted Trees
Automorphisms
AConstruction
ConstructionWordsAn Example
Lemmata and aTheorem
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 !
Elisabeth Fink Words in non-periodic branch groups
Words innon-periodic
branchgroups
ElisabethFink
Introduction
Groupsacting onRooted TreesRooted Trees
Automorphisms
AConstruction
ConstructionWordsAn Example
Lemmata and aTheorem
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 !
Elisabeth Fink Words in non-periodic branch groups
Words innon-periodic
branchgroups
ElisabethFink
Introduction
Groupsacting onRooted TreesRooted Trees
Automorphisms
AConstruction
ConstructionWordsAn Example
Lemmata and aTheorem
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 !
Elisabeth Fink Words in non-periodic branch groups
Words innon-periodic
branchgroups
ElisabethFink
Introduction
Groupsacting onRooted TreesRooted Trees
Automorphisms
AConstruction
ConstructionWordsAn Example
Lemmata and aTheorem
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 !
Elisabeth Fink Words in non-periodic branch groups
Words innon-periodic
branchgroups
ElisabethFink
Introduction
Groupsacting onRooted TreesRooted Trees
Automorphisms
AConstruction
ConstructionWordsAn Example
Lemmata and aTheorem
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
´i 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.
Elisabeth Fink Words in non-periodic branch groups
Words innon-periodic
branchgroups
ElisabethFink
Introduction
Groupsacting onRooted TreesRooted Trees
Automorphisms
AConstruction
ConstructionWordsAn Example
Lemmata and aTheorem
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
´i 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.
Elisabeth Fink Words in non-periodic branch groups
Words innon-periodic
branchgroups
ElisabethFink
Introduction
Groupsacting onRooted TreesRooted Trees
Automorphisms
AConstruction
ConstructionWordsAn Example
Lemmata and aTheorem
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
´i 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.
Elisabeth Fink Words in non-periodic branch groups
Words innon-periodic
branchgroups
ElisabethFink
Introduction
Groupsacting onRooted TreesRooted Trees
Automorphisms
AConstruction
ConstructionWordsAn Example
Lemmata and aTheorem
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
´i 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.
Elisabeth Fink Words in non-periodic branch groups
Words innon-periodic
branchgroups
ElisabethFink
Introduction
Groupsacting onRooted TreesRooted Trees
Automorphisms
AConstruction
ConstructionWordsAn Example
Lemmata and aTheorem
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
´i 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.
Elisabeth Fink Words in non-periodic branch groups
Words innon-periodic
branchgroups
ElisabethFink
Introduction
Groupsacting onRooted TreesRooted Trees
Automorphisms
AConstruction
ConstructionWordsAn Example
Lemmata and aTheorem
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
´i 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.
Elisabeth Fink Words in non-periodic branch groups
Words innon-periodic
branchgroups
ElisabethFink
Introduction
Groupsacting onRooted TreesRooted Trees
Automorphisms
AConstruction
ConstructionWordsAn Example
Lemmata and aTheorem
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
´i 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.
Elisabeth Fink Words in non-periodic branch groups
Words innon-periodic
branchgroups
ElisabethFink
Introduction
Groupsacting onRooted TreesRooted Trees
Automorphisms
AConstruction
ConstructionWordsAn Example
Lemmata and aTheorem
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
´i 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.
Elisabeth Fink Words in non-periodic branch groups
Words innon-periodic
branchgroups
ElisabethFink
Introduction
Groupsacting onRooted TreesRooted Trees
Automorphisms
AConstruction
ConstructionWordsAn Example
Lemmata and aTheorem
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
Elisabeth Fink Words in non-periodic branch groups
Words innon-periodic
branchgroups
ElisabethFink
Introduction
Groupsacting onRooted TreesRooted Trees
Automorphisms
AConstruction
ConstructionWordsAn Example
Lemmata and aTheorem
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
Elisabeth Fink Words in non-periodic branch groups
Words innon-periodic
branchgroups
ElisabethFink
Introduction
Groupsacting onRooted TreesRooted Trees
Automorphisms
AConstruction
ConstructionWordsAn Example
Lemmata and aTheorem
Growth
FurtherQuestions
Growth
Proposition
G does not have polynomial growth.
Self-Similarities
Find copies of G1 in G:
G1 = 〈a1, b1〉 G1 = 〈a1, b1〉
b1a1 a1 b1b1 b1a1
Elisabeth Fink Words in non-periodic branch groups
Words innon-periodic
branchgroups
ElisabethFink
Introduction
Groupsacting onRooted TreesRooted Trees
Automorphisms
AConstruction
ConstructionWordsAn Example
Lemmata and aTheorem
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
Elisabeth Fink Words in non-periodic branch groups
Words innon-periodic
branchgroups
ElisabethFink
Introduction
Groupsacting onRooted TreesRooted Trees
Automorphisms
AConstruction
ConstructionWordsAn Example
Lemmata and aTheorem
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
Elisabeth Fink Words in non-periodic branch groups
Words innon-periodic
branchgroups
ElisabethFink
Introduction
Groupsacting onRooted TreesRooted Trees
Automorphisms
AConstruction
ConstructionWordsAn Example
Lemmata and aTheorem
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
´i ·Qi
k=0 pk for all sufficiently large i ,then G has exponential growth.
Elisabeth Fink Words in non-periodic branch groups
Words innon-periodic
branchgroups
ElisabethFink
Introduction
Groupsacting onRooted TreesRooted Trees
Automorphisms
AConstruction
ConstructionWordsAn Example
Lemmata and aTheorem
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
´i ·Qi
k=0 pk for all sufficiently large i ,then G has exponential growth.
Elisabeth Fink Words in non-periodic branch groups
Words innon-periodic
branchgroups
ElisabethFink
Introduction
Groupsacting onRooted TreesRooted Trees
Automorphisms
AConstruction
ConstructionWordsAn Example
Lemmata and aTheorem
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
´i ·Qi
k=0 pk for all sufficiently large i ,then G has exponential growth.
Elisabeth Fink Words in non-periodic branch groups
Words innon-periodic
branchgroups
ElisabethFink
Introduction
Groupsacting onRooted TreesRooted Trees
Automorphisms
AConstruction
ConstructionWordsAn Example
Lemmata and aTheorem
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
´i ·Qi
k=0 pk for all sufficiently large i ,then G has exponential growth.
Elisabeth Fink Words in non-periodic branch groups
Words innon-periodic
branchgroups
ElisabethFink
Introduction
Groupsacting onRooted TreesRooted Trees
Automorphisms
AConstruction
ConstructionWordsAn Example
Lemmata and aTheorem
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?
Elisabeth Fink Words in non-periodic branch groups
Words innon-periodic
branchgroups
ElisabethFink
Introduction
Groupsacting onRooted TreesRooted Trees
Automorphisms
AConstruction
ConstructionWordsAn Example
Lemmata and aTheorem
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?
Elisabeth Fink Words in non-periodic branch groups
Words innon-periodic
branchgroups
ElisabethFink
Introduction
Groupsacting onRooted TreesRooted Trees
Automorphisms
AConstruction
ConstructionWordsAn Example
Lemmata and aTheorem
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?
Elisabeth Fink Words in non-periodic branch groups
Words innon-periodic
branchgroups
ElisabethFink
Introduction
Groupsacting onRooted TreesRooted Trees
Automorphisms
AConstruction
ConstructionWordsAn Example
Lemmata and aTheorem
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?
Elisabeth Fink Words in non-periodic branch groups
Words innon-periodic
branchgroups
ElisabethFink
Introduction
Groupsacting onRooted TreesRooted Trees
Automorphisms
AConstruction
ConstructionWordsAn Example
Lemmata and aTheorem
Growth
FurtherQuestions
Thank you!
Elisabeth Fink Words in non-periodic branch groups