Invariant random subgroups and applications

Invariant random subgroups and applications

Yair Glasner Department of Mathematics Ben-Gurion university of the Negev.

Plan

1 Plan

2 IRS Definitions, Examples, Theorems

3 Essential subgroups

IRS

Definition Γa l.c group.

Sub(Γ) ={subgroups ofΓ}. Compact Chaubuty topology. Γdiscrete

d(∆,Σ) =e−V V =sup R

{[∆∩B(R)] = [Σ∩B(R)]}.

Examples of IRS

Example

Normal: δN forN_CΓ.

LatticeµΓ:∆ =gΓg−1 g Haar random. Standard: w∗-limit of such.

Sofic question: Is every IRS a limit?

Universal: Γx for a p.m.p actionΓy(X,B, µ).

Induction PrnIndG_Γ(∆)∈Ao= Z G/Γ n Prγ−1∆γ ∈Aodµ(γ).

Measurable simplicity

Theorem (Stuck-Zimmer)

G - Simple Lie groupRrk(G)≥2. Non-trivial ergodic IRS: (i)δhei, (ii)δG, (iii)µΓ.

Follows for lattices by induction.

Generalizes Margulis Normal subgroup theorem. False for SL2(R).

Graph Theorey,

Xn: finite graphs(q+1)-regular,

a`(n) = #{closed non-BT paths, length`inXn}. γ`=limn→ ∞a`(n)(on subsequence).

Theorem (Abért - G - Virág) The following are equivalent

γ`=0 ∀`∈N P∞

Spectral reformulation

Xn: finite graphs(q+1)-regular,

µ(n) =Spectral measure of RW operator. µ=limn→ ∞µ(n)(on subsequence).

Theorem (Abért - G - Virág) The following are equivalent

µ([−2√q,2√q]) =1 µ=µKM = q₂+_π1

√ 4q−x2 (q+1)2_−x2dx

Spectral reformulation

Xn: finite graphs(q+1)-regular,

µ(n) =Spectral measure of RW operator. µ=limn→ ∞µ(n)(on subsequence).

Theorem (Abért - G - Virág) The following are equivalent

µ([−2√q,2√q]) =1 µ=µKM = q_2π+1

√

4q−x2

Limit multiplicity theorems

Xndifferent, compact,G-loc-sym. (GsimpleRrk≥2). X =G/K the symmetric space.

Theorem (Abért, Bergeron, Biringer, Gelander, Nikolov, Raimbault, Samet) m(π,Xn) vol(Xn) → d(π) ∀π∈ ˆ G,d(π) =formal dimension bk(Xn) vol(Xn) → β_k(2)(X) ∀k ∈N

Realization problem for Furstenberg entropy

Nevo-Zimmer: GconnectedRrkG≥2. Thenhµattains only finitely many values.

Theorem (Bowen)

Probability preserving actions

Γ =hSi<GLn(F).

No amenable normal subgroups. No commuting normal subgroups. Theorem (G)

There exits a free subgroup F <Γsuch that For every non-free p.m.p actionΓy(X,B, µ)

F ·x = Γ·x , a.s.

Tits alternative for IRS

Γ =hSi<GLn(F).

No amenable normal subgroups. No commuting normal subgroups. Theorem

There exits a free subgroup F <Γsuch that For every IRShei 6= ∆<Γ

F∆ = Γ, a.s.

Essential subgroups

Definition (Essential subgroups)

Let∆<Γbe an IRS. A subgroupM <Γisessentialif Pr{M <∆} 0

Lemma

If∆<Γis an IRS in a countable group. Then∆is a.s. locally essential.

Proof.LetΣ1,Σ2, . . .be the f.g. non-essential subgroups.

{∆∈Sub(Γ)|∆is not loc. ess.}=S

i{∆∈Sub(Γ)|∆>Σi} and the right hand side is a countable union of null sets.

Some rep

theory

We will assume from now on

Γ<GLn+1(k)withk local field. (Need something more general whenΓis not f.g.)

Γstrongly irreducible onP(kn)

H= ΓZ simple, with trivial center.

∃strongly proximalg∈Γ. Definition

Ef,SI = Stronly irreducible f.g. essential subgroups = {Σ1,Σ2, . . .}

Two central propositions

Theorem

Every non-trivial IRS∆<Γcontains someΣ∈ Ef,SI _a.s. Theorem

There exits elements:

{a(i,j, γ)∈Γ|i,j ∈N, γ∈Γ}

that

Freely generate a free group, a(i,j, γ)∈ΣiγΣj

A baby case

We will show: The IRS∆<Γa.s. contains an infinite ess. subgroup. Assume the contrary.

In fact condition on all essential subgroups being finite. ∆is locally essential a.s. ⇒locally finite a.s.

⇒∆has a finite orbitY ={s1,s2, . . . ,sl} ⊂P(kn)a.s. λ∈Γ- essential element.

g ∈Γvery proximal element

v,H attracting point and repelling hyperplane forg. w.l.o.gY ∩Hg=∅.

A baby case II

By Poincaré recurrence:

for almost all∆withλ∈∆we havegnλg−n ∈∆for infinitely manyn.

For such subsequenceλgnmY =gnmY. ButgnmY→v. Sinceλ,gare (almost) general. All essential elements are trivial.