arXiv 1407.28722014-07math.GRmath.DS

线性群的不变随机子群

Invariant random subgroups of linear groups

Tsachik Gelander, Yair Glasner

备注：: Main article by Yair Glasner with an appendix by Tsachik Gelander and Yair Glasner. 41 pages 5 figures

摘要：

设$Γ< \mathrm{GL}_n(F)$是一个可数的非顺从线性群，它具有一个简单的无中心Zebraki闭包，$\mathrm{Sub}(Γ)$是$Γ$的所有子群的空间，它具有紧的度量的Chabauty拓扑。$Γ$的不变随机子群（IRS）是$\mathrm{Sub}(Γ)$上的共轭不变Borel概率测度。一个$\mathrm{IRS}$被称为非平凡的，如果它在平凡群中没有原子，即如果它几乎必然是非平凡的。我们用$\mathrm{IRS}^{0}(Γ)$表示$Γ$上所有非平凡$\mathrm{IRS}$的集合。证明了在$Γ$上存在自由子群$F < Γ$和非离散群拓扑$\mathrm{St}$，使得对于每个$μ\in \mathrm{IRS}^{0}(Γ)$，下列性质成立：（i）$μ$ -几乎$Γ$的每个子群都是开的. (ii)$F \cdot Δ= Γ$用于$μ$ -几乎所有$Δ\in \mathrm{Sub}(Γ)$。(iii)$F \cap Δ$是无限生成的，对于每个开子群。(iv)由$Δ\mapsto Δ\cap F$给出的映射$Φ: (\mathrm{Sub}(Γ),μ) \rightarrow (\mathrm{Sub}(F),Φ_* μ)$是概率空间的$F$ -不变同构.我们说$Γ$在概率空间上的作用，通过测度保持变换，几乎必然非自由（ASNF），如果几乎所有的点稳定子都是非平凡的。作为上述定理的一个推论，我们证明了许多ANSF $Γ$ -空间与对角$Γ$作用的乘积是ASNF。设$Γ< \mathrm{GL}_n(F)$是可数线性群，$A \lhd Γ$是$Γ$的极大正规顺从子群.我们表明，如果$μ\in \mathrm{IRS}(Γ)$是支持顺从的$Γ$子群，那么事实上，它是支持$\mathrm{Sub}(A)$。特别是，如果$A(Γ) = \langle e \rangle$，则几乎肯定是$Δ= \langle e \rangle, μ$。

英文摘要：

Let $Γ< \mathrm{GL}_n(F)$ be a countable non-amenable linear group with a simple, center free Zariski closure, $\mathrm{Sub}(Γ)$ 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 $\mathrm{Sub}(Γ)$. An $\mathrm{IRS}$ is called nontrivial if it does not have an atom in the trivial group, i.e. if it is nontrivial almost surely. We denote by $\mathrm{IRS}^{0}(Γ)$ the collection of all nontrivial $\mathrm{IRS}$ on $Γ$. We show that there exits a free subgroup $F < Γ$ and a non-discrete group topology $\mathrm{St}$ on $Γ$ such that for every $μ\in \mathrm{IRS}^{0}(Γ)$ the following properties hold: (i) $μ$-almost every subgroup of $Γ$ is open. (ii) $F \cdot Δ= Γ$ for $μ$-almost every $Δ\in \mathrm{Sub}(Γ)$. (iii) $F \cap Δ$ is infinitely generated, for every open subgroup. (iv) The map $Φ: (\mathrm{Sub}(Γ),μ) \rightarrow (\mathrm{Sub}(F),Φ_* μ)$ given by $Δ\mapsto Δ\cap F$, is an $F$-invariant isomorphism of probability spaces. 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. As a corollary of the above theorem we show that the product of finitely many ANSF $Γ$-spaces, with the diagonal $Γ$ action, is ASNF. Let $Γ< \mathrm{GL}_n(F)$ be a countable linear group, $A \lhd Γ$ the maximal normal amenable subgroup of $Γ$. We show that if $μ\in \mathrm{IRS}(Γ)$ is supported on amenable subgroups of $Γ$ then in fact it is supported on $\mathrm{Sub}(A)$. In particular if $A(Γ) = \langle e \rangle$ then $Δ= \langle e \rangle, μ$ almost surely.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

相关论文