TY - JOUR

T1 - Invariant random subgroups of linear groups

AU - Glasner, Yair

N1 - Publisher Copyright:
© 2017, Hebrew University of Jerusalem.

PY - 2017/4/1

Y1 - 2017/4/1

N2 - Let Γ < GLn(F) be a countable non-amenable linear group with a simple, center free Zariski closure. Let Sub(Γ) denote the space of all subgroups of Γ with the compact, metric, Chabauty topology. An invariant random subgroup (IRS) of Γ is a conjugation invariant Borel probability measure on Sub(Γ). An IRS is called non-trivial if it does not have an atom in the trivial group, i.e. if it is non-trivial almost surely. We denote by IRS0(Γ) the collection of all non-trivial IRS on Γ. Theorem 0.1: With the above notation, there exists a free subgroupF < Γ and a non-discrete group topology on Γ such that for everyμ ∈ IRS0(Γ) the following properties hold:μ-almost every subgroup of Γ is openF ·Δ = Γ for μ-almost every Δ ∈ Sub(Γ).F ∩ Δ is infinitely generated, for every open subgroup. In particular, this holds for μ-almost every Δ ∈ Sub(Γ).The map Φ: (Sub(Γ), μ) → (Sub(F),Φ*μ) Δ → Δ ∩ Fis an F-invariant isomorphism of probability spaces. A more technical version of this theorem is valid for general countable linear groups. We say that an action of Γ on a probability space, by measure preserving transformations, is almost surely non-free (ASNF) if almost all point stabilizers are non-trivial. Corollary 0.2: Let Γ be as in the Theorem above. Then the product of finitely many ASNF Γ-spaces, with the diagonal Γ action, is ASNF. Corollary 0.3: Let Γ < GLn(F) be a countable linear group, A Δ Γ the maximal normal amenable subgroup of Γ — its amenable radical. If μ ∈ IRS(Γ) is supported on amenable subgroups of Γ, then in fact it is supported on Sub(A). In particular, if A(Γ) = then Δ = , μ almost surely.

AB - Let Γ < GLn(F) be a countable non-amenable linear group with a simple, center free Zariski closure. Let Sub(Γ) denote the space of all subgroups of Γ with the compact, metric, Chabauty topology. An invariant random subgroup (IRS) of Γ is a conjugation invariant Borel probability measure on Sub(Γ). An IRS is called non-trivial if it does not have an atom in the trivial group, i.e. if it is non-trivial almost surely. We denote by IRS0(Γ) the collection of all non-trivial IRS on Γ. Theorem 0.1: With the above notation, there exists a free subgroupF < Γ and a non-discrete group topology on Γ such that for everyμ ∈ IRS0(Γ) the following properties hold:μ-almost every subgroup of Γ is openF ·Δ = Γ for μ-almost every Δ ∈ Sub(Γ).F ∩ Δ is infinitely generated, for every open subgroup. In particular, this holds for μ-almost every Δ ∈ Sub(Γ).The map Φ: (Sub(Γ), μ) → (Sub(F),Φ*μ) Δ → Δ ∩ Fis an F-invariant isomorphism of probability spaces. A more technical version of this theorem is valid for general countable linear groups. We say that an action of Γ on a probability space, by measure preserving transformations, is almost surely non-free (ASNF) if almost all point stabilizers are non-trivial. Corollary 0.2: Let Γ be as in the Theorem above. Then the product of finitely many ASNF Γ-spaces, with the diagonal Γ action, is ASNF. Corollary 0.3: Let Γ < GLn(F) be a countable linear group, A Δ Γ the maximal normal amenable subgroup of Γ — its amenable radical. If μ ∈ IRS(Γ) is supported on amenable subgroups of Γ, then in fact it is supported on Sub(A). In particular, if A(Γ) = then Δ = , μ almost surely.

UR - http://www.scopus.com/inward/record.url?scp=85018307527&partnerID=8YFLogxK

U2 - 10.1007/s11856-017-1479-x

DO - 10.1007/s11856-017-1479-x

M3 - Article

AN - SCOPUS:85018307527

VL - 219

SP - 215

EP - 270

JO - Israel Journal of Mathematics

JF - Israel Journal of Mathematics

SN - 0021-2172

IS - 1

ER -