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Representation Statements Groups Dynamics Metric Application
Infinite Primitive Permutation Groups.
Yair Glasner (Joint with Tsachik Gelander)
School of MathematicsInstitute for advanced study.
Texas A&M, January 2006
Representation Statements Groups Dynamics Metric Application
Outline
1 Representation theories
2 Statements of main theorems.
3 Group theoretic part of the proof.
4 Dynamics on the boundary
5 Accessing infinite index subgroups.
6 Application - Frattini Subgroups
Representation Statements Groups Dynamics Metric Application
We study groups through their actions.
Finitely generated groups:Linear groups:
Tits alternative,either contains a free group, or virtually solvable,Residually finite,Zariski topology,
Hyperbolic groups:Tits alternative,many quotients,
Permutation groups.Any group is a permutation group,
Representation Statements Groups Dynamics Metric Application
We study groups through their actions.
Finitely generated groups:Linear groups:
Tits alternative,either contains a free group, or virtually solvable,Residually finite,Zariski topology,
Hyperbolic groups:Tits alternative,many quotients,
Permutation groups.Any group is a permutation group,
Representation Statements Groups Dynamics Metric Application
We study groups through their actions.
Finitely generated groups:Linear groups:
Tits alternative,either contains a free group, or virtually solvable,Residually finite,Zariski topology,
Hyperbolic groups:Tits alternative,many quotients,
Permutation groups.Any group is a permutation group,
Representation Statements Groups Dynamics Metric Application
We study groups through their actions.
Finitely generated groups:Linear groups:
Tits alternative,either contains a free group, or virtually solvable,Residually finite,Zariski topology,
Hyperbolic groups:Tits alternative,many quotients,
Permutation groups.Any group is a permutation group,
Representation Statements Groups Dynamics Metric Application
We study groups through their actions.
Finitely generated groups:Linear groups:
Tits alternative,either contains a free group, or virtually solvable,Residually finite,Zariski topology,
Hyperbolic groups:Tits alternative,many quotients,
Permutation groups.Any group is a permutation group,
Representation Statements Groups Dynamics Metric Application
We study groups through their actions.
Finitely generated groups:Linear groups:
Tits alternative,either contains a free group, or virtually solvable,Residually finite,Zariski topology,
Hyperbolic groups:Tits alternative,many quotients,
Permutation groups.Any group is a permutation group,
Representation Statements Groups Dynamics Metric Application
We study groups through their actions.
Finitely generated groups:Linear groups:
Tits alternative,either contains a free group, or virtually solvable,Residually finite,Zariski topology,
Hyperbolic groups:Tits alternative,many quotients,
Permutation groups.Any group is a permutation group,
Representation Statements Groups Dynamics Metric Application
We study groups through their actions.
Finitely generated groups:Linear groups:
Tits alternative,either contains a free group, or virtually solvable,Residually finite,Zariski topology,
Hyperbolic groups:Tits alternative,many quotients,
Permutation groups.Any group is a permutation group,
Representation Statements Groups Dynamics Metric Application
Decompositions.
Orbit decomposition. ⇒ Transitive actions Γ/∆.
Factors (invariant equivalence relations).A group action is called primitive if
No factors.Γ Γ/∆, where ∆ < Γ is maximal.
A group is called primitive if it admits a faithful primitiveaction.
Representation Statements Groups Dynamics Metric Application
Decompositions.
Orbit decomposition. ⇒ Transitive actions Γ/∆.
Factors (invariant equivalence relations).A group action is called primitive if
No factors.Γ Γ/∆, where ∆ < Γ is maximal.
A group is called primitive if it admits a faithful primitiveaction.
Representation Statements Groups Dynamics Metric Application
Decompositions.
Orbit decomposition. ⇒ Transitive actions Γ/∆.
Factors (invariant equivalence relations).A group action is called primitive if
No factors.Γ Γ/∆, where ∆ < Γ is maximal.
A group is called primitive if it admits a faithful primitiveaction.
Representation Statements Groups Dynamics Metric Application
Decompositions.
Orbit decomposition. ⇒ Transitive actions Γ/∆.
Factors (invariant equivalence relations).A group action is called primitive if
No factors.Γ Γ/∆, where ∆ < Γ is maximal.
A group is called primitive if it admits a faithful primitiveaction.
Representation Statements Groups Dynamics Metric Application
Decompositions.
Orbit decomposition. ⇒ Transitive actions Γ/∆.
Factors (invariant equivalence relations).A group action is called primitive if
No factors.Γ Γ/∆, where ∆ < Γ is maximal.
A group is called primitive if it admits a faithful primitiveaction.
Representation Statements Groups Dynamics Metric Application
Decompositions.
Orbit decomposition. ⇒ Transitive actions Γ/∆.
Factors (invariant equivalence relations).A group action is called primitive if
No factors.Γ Γ/∆, where ∆ < Γ is maximal.
A group is called primitive if it admits a faithful primitiveaction.
Representation Statements Groups Dynamics Metric Application
Decompositions.
Orbit decomposition. ⇒ Transitive actions Γ/∆.
Factors (invariant equivalence relations).A group action is called primitive if
No factors.Γ Γ/∆, where ∆ < Γ is maximal.
A group is called primitive if it admits a faithful primitiveaction.
Representation Statements Groups Dynamics Metric Application
Primitive Groups.
Basic question
Understand primitive groups.
Similar questions.
Which groups admit a faithful ....
irreducible unitary representation?
ergodic measure preserving action?
Representation Statements Groups Dynamics Metric Application
Primitive Groups.
Basic question
Understand primitive groups.
Similar questions.
Which groups admit a faithful ....
irreducible unitary representation?
ergodic measure preserving action?
Representation Statements Groups Dynamics Metric Application
Primitive Groups.
Basic question
Understand primitive groups.
Similar questions.
Which groups admit a faithful ....
irreducible unitary representation?
ergodic measure preserving action?
Representation Statements Groups Dynamics Metric Application
Some answers
Group = (finitely generated) + (no finite normal subgroup).
Theorem (Imprecise version)
Linear group is primitive ⇔ ∃ linear rep with simple Zariskiclosure.
Theorem
Γ < Mod(S) is primitive ⇔ Irreducible and not virtually cyclic.
TheoremA hyperbolic group is primitive ⇔ is not virtually cyclic.
Theorem
A group acting minimally faithfully on a tree is always primitive.
Representation Statements Groups Dynamics Metric Application
Some answers
Group = (finitely generated) + (no finite normal subgroup).
Theorem (Imprecise version)
Linear group is primitive ⇔ ∃ linear rep with simple Zariskiclosure.
Theorem
Γ < Mod(S) is primitive ⇔ Irreducible and not virtually cyclic.
TheoremA hyperbolic group is primitive ⇔ is not virtually cyclic.
Theorem
A group acting minimally faithfully on a tree is always primitive.
Representation Statements Groups Dynamics Metric Application
Some answers
Group = (finitely generated) + (no finite normal subgroup).
Theorem (Imprecise version)
Linear group is primitive ⇔ ∃ linear rep with simple Zariskiclosure.
Theorem
Γ < Mod(S) is primitive ⇔ Irreducible and not virtually cyclic.
TheoremA hyperbolic group is primitive ⇔ is not virtually cyclic.
Theorem
A group acting minimally faithfully on a tree is always primitive.
Representation Statements Groups Dynamics Metric Application
Some answers
Group = (finitely generated) + (no finite normal subgroup).
Theorem (Imprecise version)
Linear group is primitive ⇔ ∃ linear rep with simple Zariskiclosure.
Theorem
Γ < Mod(S) is primitive ⇔ Irreducible and not virtually cyclic.
TheoremA hyperbolic group is primitive ⇔ is not virtually cyclic.
Theorem
A group acting minimally faithfully on a tree is always primitive.
Representation Statements Groups Dynamics Metric Application
Some answers
Group = (finitely generated) + (no finite normal subgroup).
Theorem (Imprecise version)
Linear group is primitive ⇔ ∃ linear rep with simple Zariskiclosure.
Theorem
Γ < Mod(S) is primitive ⇔ Irreducible and not virtually cyclic.
TheoremA hyperbolic group is primitive ⇔ is not virtually cyclic.
Theorem
A group acting minimally faithfully on a tree is always primitive.
Representation Statements Groups Dynamics Metric Application
Some answers
Group = (finitely generated) + (no finite normal subgroup).
Theorem (Imprecise version)
Linear group is primitive ⇔ ∃ linear rep with simple Zariskiclosure.
Theorem
Γ < Mod(S) is primitive ⇔ Irreducible and not virtually cyclic.
TheoremA hyperbolic group is primitive ⇔ is not virtually cyclic.
Theorem
A group acting minimally faithfully on a tree is always primitive.
Representation Statements Groups Dynamics Metric Application
Some answers
Group = (finitely generated) + (no finite normal subgroup).
Theorem (Imprecise version)
Linear group is primitive ⇔ ∃ linear rep with simple Zariskiclosure.
Theorem
Γ < Mod(S) is primitive ⇔ Irreducible and not virtually cyclic.
TheoremA hyperbolic group is primitive ⇔ is not virtually cyclic.
Theorem
A group acting minimally faithfully on a tree is always primitive.
Representation Statements Groups Dynamics Metric Application
Some answers
Group = (finitely generated) + (no finite normal subgroup).
Theorem (Imprecise version)
Linear group is primitive ⇔ ∃ linear rep with simple Zariskiclosure.
Theorem
Γ < Mod(S) is primitive ⇔ Irreducible and not virtually cyclic.
TheoremA hyperbolic group is primitive ⇔ is not virtually cyclic.
Theorem
A group acting minimally faithfully on a tree is always primitive.
Representation Statements Groups Dynamics Metric Application
Precise version of main theorem
Theorem
A finitely generated linear group Γ is primitive if and only if thereexists a linear representation Γ < GLn(k) over an algebraically
closed field, with Zariski closure G = ΓZ
such that,
G0 = H × H × . . .× H a product of simple groups.
Γ acts faithfully and transitively on the H’s.
Representation Statements Groups Dynamics Metric Application
countable groups
TheoremLet Γ be a non-torsion, countable quasi-primitive linear group.Then Γ one of the following:
Simple closure: Γ has “Simple Zariski closure” as above.
Affine Γ = ∆ n F n, where F is a prime field 1 ≤ n ≤ ∞ and∆ < GLn(F ) acts without invariant subgroups. E.g. Q∗ n Q
Diagonal Γ = ∆ n H, where H is a nonabeliancharacteristically simple group and ∆ acts with no invariantsubgroups.
In the affine and diagonal case the quasi-primitive action isprimitive and unique.
Example
PSLn(Fp) is a torsion primitive group. The group
Representation Statements Groups Dynamics Metric Application
countable groups
TheoremLet Γ be a non-torsion, countable quasi-primitive linear group.Then Γ one of the following:
Simple closure: Γ has “Simple Zariski closure” as above.
Affine Γ = ∆ n F n, where F is a prime field 1 ≤ n ≤ ∞ and∆ < GLn(F ) acts without invariant subgroups. E.g. Q∗ n Q
Diagonal Γ = ∆ n H, where H is a nonabeliancharacteristically simple group and ∆ acts with no invariantsubgroups.
In the affine and diagonal case the quasi-primitive action isprimitive and unique.
Example
PSLn(Fp) is a torsion primitive group. The group
Representation Statements Groups Dynamics Metric Application
Margulis Soı̆fer
Let Γ be a finitely generated linear group.Our work
Theorem
Γ admits a faithful primitive action ⇔ has simple Zariski closure.
is inspired by the following:
Theorem (Margulis Soı̆fer)
Γ admits an infinite primitive action ⇔ not virtually solvable.
Which in turn was inspired by:
Theorem (Tits alternative)
Γ contains a non-abelian free subgroup ⇔ not virtually solvable.
Representation Statements Groups Dynamics Metric Application
Margulis Soı̆fer
Let Γ be a finitely generated linear group.Our work
Theorem
Γ admits a faithful primitive action ⇔ has simple Zariski closure.
is inspired by the following:
Theorem (Margulis Soı̆fer)
Γ admits an infinite primitive action ⇔ not virtually solvable.
Which in turn was inspired by:
Theorem (Tits alternative)
Γ contains a non-abelian free subgroup ⇔ not virtually solvable.
Representation Statements Groups Dynamics Metric Application
Margulis Soı̆fer
Let Γ be a finitely generated linear group.Our work
Theorem
Γ admits a faithful primitive action ⇔ has simple Zariski closure.
is inspired by the following:
Theorem (Margulis Soı̆fer)
Γ admits an infinite primitive action ⇔ not virtually solvable.
Which in turn was inspired by:
Theorem (Tits alternative)
Γ contains a non-abelian free subgroup ⇔ not virtually solvable.
Representation Statements Groups Dynamics Metric Application
Margulis Soı̆fer
Let Γ be a finitely generated linear group.Our work
Theorem
Γ admits a faithful primitive action ⇔ has simple Zariski closure.
is inspired by the following:
Theorem (Margulis Soı̆fer)
Γ admits an infinite primitive action ⇔ not virtually solvable.
Which in turn was inspired by:
Theorem (Tits alternative)
Γ contains a non-abelian free subgroup ⇔ not virtually solvable.
Representation Statements Groups Dynamics Metric Application
Margulis Soı̆fer
Let Γ be a finitely generated linear group.Our work
Theorem
Γ admits a faithful primitive action ⇔ has simple Zariski closure.
is inspired by the following:
Theorem (Margulis Soı̆fer)
Γ admits an infinite primitive action ⇔ not virtually solvable.
Which in turn was inspired by:
Theorem (Tits alternative)
Γ contains a non-abelian free subgroup ⇔ not virtually solvable.
Representation Statements Groups Dynamics Metric Application
Margulis Soı̆fer
Let Γ be a finitely generated linear group.Our work
Theorem
Γ admits a faithful primitive action ⇔ has simple Zariski closure.
is inspired by the following:
Theorem (Margulis Soı̆fer)
Γ admits an infinite primitive action ⇔ not virtually solvable.
Which in turn was inspired by:
Theorem (Tits alternative)
Γ contains a non-abelian free subgroup ⇔ not virtually solvable.
Representation Statements Groups Dynamics Metric Application
Margulis Soı̆fer
Let Γ be a finitely generated linear group.Our work
Theorem
Γ admits a faithful primitive action ⇔ has simple Zariski closure.
is inspired by the following:
Theorem (Margulis Soı̆fer)
Γ admits an infinite primitive action ⇔ not virtually solvable.
Which in turn was inspired by:
Theorem (Tits alternative)
Γ contains a non-abelian free subgroup ⇔ not virtually solvable.
Representation Statements Groups Dynamics Metric Application
Strategy.
Margulis Soı̆fer Our work
Have: Γ not virt-solvable simple Zariski closure.Want: M < Γ maximal of ∞-index maximal with trivial core.How? ∆ < Γ profinitely dense pro dense
Γ // // G
M?�
OO∃
>> >>~~
~~
∆?�
OO∃
GG GG��
��
��
�
Note:Any subgroup in contained in a maximal subgroup. The onlyproblem is it might be too large.
Representation Statements Groups Dynamics Metric Application
Strategy.
Margulis Soı̆fer Our work
Have: Γ not virt-solvable simple Zariski closure.Want: M < Γ maximal of ∞-index maximal with trivial core.How? ∆ < Γ profinitely dense pro dense
Γ // // G
M?�
OO∃
>> >>~~
~~
∆?�
OO∃
GG GG��
��
��
�
Note:Any subgroup in contained in a maximal subgroup. The onlyproblem is it might be too large.
Representation Statements Groups Dynamics Metric Application
Strategy.
Margulis Soı̆fer Our work
Have: Γ not virt-solvable simple Zariski closure.Want: M < Γ maximal of ∞-index maximal with trivial core.How? ∆ < Γ profinitely dense pro dense
Γ // // G
M?�
OO∃
>> >>~~
~~
∆?�
OO∃
GG GG��
��
��
�
Note:Any subgroup in contained in a maximal subgroup. The onlyproblem is it might be too large.
Representation Statements Groups Dynamics Metric Application
Strategy.
Margulis Soı̆fer Our work
Have: Γ not virt-solvable simple Zariski closure.Want: M < Γ maximal of ∞-index maximal with trivial core.How? ∆ < Γ profinitely dense pro dense
Γ // // G
M?�
OO∃
>> >>~~
~~
∆?�
OO∃
GG GG��
��
��
�
Note:Any subgroup in contained in a maximal subgroup. The onlyproblem is it might be too large.
Representation Statements Groups Dynamics Metric Application
Prodense subgroups
Definition (Prodense subgroups)
A prodense subgroup ∆ < Γ is one that maps onto everyproper quotient Γ/N of Γ.
proposition
A finitely generated group Γ is prodense if and only if it containsa proper prodense subgroup.
Theorem (Abert-G)
Let Γ < G be a dense subgroup is a totally disconnected simplegroup. And let ∆ < Γ be a relatively open subgroup. Then ∆ isprodense.E.g G = PGLn(Qp), Γ = PGLn(Z[1/p]),∆ = PGLn(Z).
Representation Statements Groups Dynamics Metric Application
Prodense subgroups
Definition (Prodense subgroups)
A prodense subgroup ∆ < Γ is one that maps onto everyproper quotient Γ/N of Γ.
proposition
A finitely generated group Γ is prodense if and only if it containsa proper prodense subgroup.
Theorem (Abert-G)
Let Γ < G be a dense subgroup is a totally disconnected simplegroup. And let ∆ < Γ be a relatively open subgroup. Then ∆ isprodense.E.g G = PGLn(Qp), Γ = PGLn(Z[1/p]),∆ = PGLn(Z).
Representation Statements Groups Dynamics Metric Application
Free subgroups.
Question
Why free subgroups?
To make sure that ∆ 6= Γ.
Generator in each coset of each normal subgroup.
Uncountable number of normal subgroups?
Representation Statements Groups Dynamics Metric Application
Free subgroups.
Question
Why free subgroups?
To make sure that ∆ 6= Γ.
Generator in each coset of each normal subgroup.
Uncountable number of normal subgroups?
Representation Statements Groups Dynamics Metric Application
Free subgroups.
Question
Why free subgroups?
To make sure that ∆ 6= Γ.
Generator in each coset of each normal subgroup.
Uncountable number of normal subgroups?
Representation Statements Groups Dynamics Metric Application
Free subgroups.
Question
Why free subgroups?
To make sure that ∆ 6= Γ.
Generator in each coset of each normal subgroup.
Uncountable number of normal subgroups?
Representation Statements Groups Dynamics Metric Application
Contracting elements.
Definition
φ ∈ Homeo(M) is contracting if φ(M \ R) ⊂ A.A, R open & disjoint are the attracting and repellingneighborhoods.
Representation Statements Groups Dynamics Metric Application
Contracting elements.
Definition
φ ∈ Homeo(M) is contracting if φ(M \ R) ⊂ A.A, R open & disjoint are the attracting and repellingneighborhoods.
Representation Statements Groups Dynamics Metric Application
Contracting elements.
Definition
φ ∈ Homeo(M) is contracting if φ(M \ R) ⊂ A.A, R open & disjoint are the attracting and repellingneighborhoods.
Representation Statements Groups Dynamics Metric Application
Contracting elements.
Definition
φ ∈ Homeo(M) is contracting if φ(M \ R) ⊂ A.A, R open & disjoint are the attracting and repellingneighborhoods.
Representation Statements Groups Dynamics Metric Application
Contracting elements.
Definition
φ ∈ Homeo(M) is contracting if φ(M \ R) ⊂ A.A, R open & disjoint are the attracting and repellingneighborhoods.
Representation Statements Groups Dynamics Metric Application
Contracting elements.
Definition
φ ∈ Homeo(M) is contracting if φ(M \ R) ⊂ A.A, R open & disjoint are the attracting and repellingneighborhoods.
Representation Statements Groups Dynamics Metric Application
Contracting elements.
Definition
φ ∈ Homeo(M) is contracting if φ(M \ R) ⊂ A.A, R open & disjoint are the attracting and repellingneighborhoods.
Representation Statements Groups Dynamics Metric Application
Ping pong lemma
Lemma (ping-pong lemma)
Contracting homeomorphisms with disjoint neighborhoodsgenerate a free group.
Representation Statements Groups Dynamics Metric Application
Proximal elements
Definition
g ∈ Homeo(M) is proximal with attracting and repelling points a, r ,if for any open neighborhoods a ∈ A, r ∈ R some power gn iscontracting.
Representation Statements Groups Dynamics Metric Application
Proximal elements
Definition
g ∈ Homeo(M) is proximal with attracting and repelling points a, r ,if for any open neighborhoods a ∈ A, r ∈ R some power gn iscontracting.
Representation Statements Groups Dynamics Metric Application
Proximal elements
Definition
g ∈ Homeo(M) is proximal with attracting and repelling points a, r ,if for any open neighborhoods a ∈ A, r ∈ R some power gn iscontracting.
Representation Statements Groups Dynamics Metric Application
Proximal elements
Definition
g ∈ Homeo(M) is proximal with attracting and repelling points a, r ,if for any open neighborhoods a ∈ A, r ∈ R some power gn iscontracting.
Representation Statements Groups Dynamics Metric Application
Proximal elements
Definition
g ∈ Homeo(M) is proximal with attracting and repelling points a, r ,if for any open neighborhoods a ∈ A, r ∈ R some power gn iscontracting.
Representation Statements Groups Dynamics Metric Application
Proximal elements
Definition
g ∈ Homeo(M) is proximal with attracting and repelling points a, r ,if for any open neighborhoods a ∈ A, r ∈ R some power gn iscontracting.
Representation Statements Groups Dynamics Metric Application
The big ping-pong table
A proximal in every normal subgroup. Fix one proximal g ∈ Γ.
Representation Statements Groups Dynamics Metric Application
The big ping-pong table
Order (a basis for) normal subgroups N1, N2, N3, . . ..
Representation Statements Groups Dynamics Metric Application
The big ping-pong table
Induction ai ∈ Ni and gn play ping-pong.
Representation Statements Groups Dynamics Metric Application
The big ping-pong table
A proximal element of N2, not satisfying the ping-pong.
Representation Statements Groups Dynamics Metric Application
The big ping-pong table
Conjugate it by a high power of g
Representation Statements Groups Dynamics Metric Application
The big ping-pong table
Replace g 7→ gn. Here {a1, a2, gn} play Ping-Pong
Representation Statements Groups Dynamics Metric Application
The big ping-pong table
Here {a1, a2, a3, gn} play Ping-Pong.
Representation Statements Groups Dynamics Metric Application
The big ping-pong table
Proximal elements from the cosets of N1.
Representation Statements Groups Dynamics Metric Application
The big ping-pong table
Proximal elements from the cosets of N1.
Representation Statements Groups Dynamics Metric Application
The big ping-pong table
Cosets of all other normal subgroups.
Representation Statements Groups Dynamics Metric Application
Ingredients for the dynamical argument
Need large normal subgroups,
Need normal subgroups that contain proximal elements,
First step is achieved by representation theoretic tools,
Finite index problems,
not finitely generated problems,
Theorem (Tits, Margulis-Soı̆fer, Breuillard-Gelander)
Assume (ΓZ)0 is not solvable, then exists a projective strongly
irreducible representation over some local field, with a highlyproximal elements.
Representation Statements Groups Dynamics Metric Application
Ingredients for the dynamical argument
Need large normal subgroups,
Need normal subgroups that contain proximal elements,
First step is achieved by representation theoretic tools,
Finite index problems,
not finitely generated problems,
Theorem (Tits, Margulis-Soı̆fer, Breuillard-Gelander)
Assume (ΓZ)0 is not solvable, then exists a projective strongly
irreducible representation over some local field, with a highlyproximal elements.
Representation Statements Groups Dynamics Metric Application
Ingredients for the dynamical argument
Need large normal subgroups,
Need normal subgroups that contain proximal elements,
First step is achieved by representation theoretic tools,
Finite index problems,
not finitely generated problems,
Theorem (Tits, Margulis-Soı̆fer, Breuillard-Gelander)
Assume (ΓZ)0 is not solvable, then exists a projective strongly
irreducible representation over some local field, with a highlyproximal elements.
Representation Statements Groups Dynamics Metric Application
Ingredients for the dynamical argument
Need large normal subgroups,
Need normal subgroups that contain proximal elements,
First step is achieved by representation theoretic tools,
Finite index problems,
not finitely generated problems,
Theorem (Tits, Margulis-Soı̆fer, Breuillard-Gelander)
Assume (ΓZ)0 is not solvable, then exists a projective strongly
irreducible representation over some local field, with a highlyproximal elements.
Representation Statements Groups Dynamics Metric Application
Ingredients for the dynamical argument
Need large normal subgroups,
Need normal subgroups that contain proximal elements,
First step is achieved by representation theoretic tools,
Finite index problems,
not finitely generated problems,
Theorem (Tits, Margulis-Soı̆fer, Breuillard-Gelander)
Assume (ΓZ)0 is not solvable, then exists a projective strongly
irreducible representation over some local field, with a highlyproximal elements.
Representation Statements Groups Dynamics Metric Application
Ingredients for the dynamical argument
Need large normal subgroups,
Need normal subgroups that contain proximal elements,
First step is achieved by representation theoretic tools,
Finite index problems,
not finitely generated problems,
Theorem (Tits, Margulis-Soı̆fer, Breuillard-Gelander)
Assume (ΓZ)0 is not solvable, then exists a projective strongly
irreducible representation over some local field, with a highlyproximal elements.
Representation Statements Groups Dynamics Metric Application
Lipschitz = contraction
Theorem
A projective transformation is contracting, if and only if it isLipschitz on some open neighborhood.
Quantitative estimates.
Representation Statements Groups Dynamics Metric Application
Lipschitz = contraction
Theorem
A projective transformation is contracting, if and only if it isLipschitz on some open neighborhood.
Quantitative estimates.
Representation Statements Groups Dynamics Metric Application
Obtaining Lipschitz transformations
Representation Statements Groups Dynamics Metric Application
Obtaining Lipschitz transformations
Representation Statements Groups Dynamics Metric Application
Obtaining Lipschitz transformations
Representation Statements Groups Dynamics Metric Application
Obtaining Lipschitz transformations
Representation Statements Groups Dynamics Metric Application
Obtaining Lipschitz transformations
Representation Statements Groups Dynamics Metric Application
Frattini Subgroups
Definition
The Frattini subgroup φ(G) of a group G is
The intersection of all maximal subgroups.
The subgroup of all “non-generators”.
Lemma
If Γ is primitive then φ(G) = 〈e〉.
Representation Statements Groups Dynamics Metric Application
Frattini Subgroups
Definition
The Frattini subgroup φ(G) of a group G is
The intersection of all maximal subgroups.
The subgroup of all “non-generators”.
Lemma
If Γ is primitive then φ(G) = 〈e〉.
Representation Statements Groups Dynamics Metric Application
Frattini Subgroups
Definition
The Frattini subgroup φ(G) of a group G is
The intersection of all maximal subgroups.
The subgroup of all “non-generators”.
Lemma
If Γ is primitive then φ(G) = 〈e〉.
Representation Statements Groups Dynamics Metric Application
Computations of Frattini Subgroups.
Theorem (Frattini subgroups)
We compute Frattini subgroups in all geometric settings.
Linear groups (Platonov [66], Wehrfritz [68]),
Mapping class groups (Ivanov [92]),
Hyperbolic groups (I. Kapovich [03]),
Trees ⇒ Answers a question of Higman and Neumann [54](f.g. case),
Representation Statements Groups Dynamics Metric Application
The Higman Neumann question
Theorem (Conj. Higman and Neumann 54)
Let G = A ∗C B be a finitely generated amalgamated freeproduct, then φ(G) < C.
Proof.
f : G → Aut(T ), Bass-Serre Tree.
By Main Theorem φ(f (G)) = 〈e〉.φ(G) < ker(f ) < C
Representation Statements Groups Dynamics Metric Application
The Higman Neumann question
Theorem (Conj. Higman and Neumann 54)
Let G = A ∗C B be a finitely generated amalgamated freeproduct, then φ(G) < C.
Proof.
f : G → Aut(T ), Bass-Serre Tree.
By Main Theorem φ(f (G)) = 〈e〉.φ(G) < ker(f ) < C
Representation Statements Groups Dynamics Metric Application
The Higman Neumann question
Theorem (Conj. Higman and Neumann 54)
Let G = A ∗C B be a finitely generated amalgamated freeproduct, then φ(G) < C.
Proof.
f : G → Aut(T ), Bass-Serre Tree.
By Main Theorem φ(f (G)) = 〈e〉.φ(G) < ker(f ) < C
Representation Statements Groups Dynamics Metric Application
The Higman Neumann question
Theorem (Conj. Higman and Neumann 54)
Let G = A ∗C B be a finitely generated amalgamated freeproduct, then φ(G) < C.
Proof.
f : G → Aut(T ), Bass-Serre Tree.
By Main Theorem φ(f (G)) = 〈e〉.φ(G) < ker(f ) < C
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