Infinite Primitive Permutation Groups.yairgl/Texas.pdfFinitely generated groups: Linear groups: Tits alternative, either contains a free group, or virtually solvable, Residually ﬁnite,

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


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


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,












































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.


Decompositions.






Decompositions.






Decompositions.






Decompositions.






Decompositions.






Decompositions.






Primitive Groups.

Basic question

Understand primitive groups.

Similar questions.

Which groups admit a faithful ....

irreducible unitary representation?

ergodic measure preserving action?


Primitive Groups.

Basic question


Similar questions.





Primitive Groups.

Basic question


Similar questions.





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.


Some answers




Theorem



Theorem



Some answers




Theorem



Theorem



Some answers




Theorem



Theorem



Some answers




Theorem



Theorem



Some answers




Theorem



Theorem



Some answers




Theorem



Theorem



Some answers




Theorem



Theorem



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.


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


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


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.


Margulis Soı̆fer


Theorem









Margulis Soı̆fer


Theorem









Margulis Soı̆fer


Theorem









Margulis Soı̆fer


Theorem









Margulis Soı̆fer


Theorem









Margulis Soı̆fer


Theorem









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

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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).


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).


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?


Free subgroups.

Question

Why free subgroups?





Free subgroups.

Question

Why free subgroups?





Free subgroups.

Question

Why free subgroups?





Contracting elements.

Definition

φ ∈ Homeo(M) is contracting if φ(M \ R) ⊂ A.A, R open & disjoint are the attracting and repellingneighborhoods.



Definition




Definition




Definition




Definition




Definition




Definition



Ping pong lemma

Lemma (ping-pong lemma)

Contracting homeomorphisms with disjoint neighborhoodsgenerate a free group.


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.


Proximal elements

Definition



Proximal elements

Definition



Proximal elements

Definition



Proximal elements

Definition



Proximal elements

Definition



The big ping-pong table

A proximal in every normal subgroup. Fix one proximal g ∈ Γ.



Order (a basis for) normal subgroups N1, N2, N3, . . ..



Induction ai ∈ Ni and gn play ping-pong.



A proximal element of N2, not satisfying the ping-pong.



Conjugate it by a high power of g



Replace g 7→ gn. Here {a1, a2, gn} play Ping-Pong



Here {a1, a2, a3, gn} play Ping-Pong.



Proximal elements from the cosets of N1.



Proximal elements from the cosets of N1.



Cosets of all other normal subgroups.


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.




















































Lipschitz = contraction

Theorem

A projective transformation is contracting, if and only if it isLipschitz on some open neighborhood.

Quantitative estimates.


Lipschitz = contraction

Theorem

A projective transformation is contracting, if and only if it isLipschitz on some open neighborhood.

Quantitative estimates.


Obtaining Lipschitz transformations










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〉.


Frattini Subgroups

Definition




Lemma



Frattini Subgroups

Definition




Lemma



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),


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





Proof.







Proof.







Proof.



Documents

Infinite Primitive Permutation Groups.yairgl/Texas.pdfFinitely generated groups: Linear groups: Tits alternative, either contains a free group, or virtually solvable, Residually ﬁnite,