Rigidity Theorems For Actions Of Product Groups And Countable Borel Equivalence Relations
Rigidity Theorems For Actions Of Product Groups And Countable Borel Equivalence Relations
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DOI：
10.1090/memo/0833
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发表时间：
2005-07
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通讯作者：
G. Hjorth;A. Kechris
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文献类型：
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作者：
G. Hjorth;A. Kechris
这本回忆录既是对Borel等价关系理论的贡献，被认为是Borel约化，也是保持群作用的度量，被认为是直到轨道等价的。这里称E为Borel可约化为F的当且仅当f(X)Ff(Y)。此外，如果具有由等价关系提供的额外结构的相应度量空间几乎处处同构，则E是轨道等价于F。我们考虑乘积群在标准Borel概率空间上的遍历作用和保测变换。总而言之，这本专著的基本部分表明，如果所涉及的群有一个合适的“边界”概念(我们用准双曲线的定义使之精确)，那么只有当乘积群和作用量的耦合之间存在某种代数相似时，才能将一个轨道等价关系归结为另一个轨道等价关系。这对轨道等价性也有影响。在原等价关系不存在非平凡几乎不变集的情况下，该方法可以得到相对遍历的结果。一个等价关系E称为对F相对遍历的，如果任一个f与XEY⇒f(X)Ff(Y)几乎处处都有[f(X)]F常数。引理和结构定理的基本集合以多种不同的方式使用。最紧迫的问题之一是给出以前只能用齐默尔的超刚性理论得到的结果的完全自足的证明。我们给出了“初等证明”，证明了存在不可比的可数Borel等价关系(Adams-Kechris)，包含并不意味着可约(Adams)，(n+1)E不一定可约为Ne(Thomas)。在本文的后面部分，我们给出了该理论在产品群的具体情况下的应用。特别地，我们分类了自由群的乘积的作用，得到了附加的刚性定理和在此背景下的相对遍历性结果。有一系列相当长的附录，其主要目的是让读者对基本技术有一个全面的描述。但这里也包含了一些新的结果。例如，我们证明了关于来自顺从群的余圈的Furstberg-Zimmer引理关于Baire范畴是不成立的，并用这个引理回答了Weiss的一个问题。我们还给出了F2具有Haagerup逼近性质的另一种证明。
This Memoir is both a contribution to the theory of Borel equivalence relations, considered up to Borel reducibility, and measure preserving group actions considered up to orbit equivalence. Here E is said to be Borel reducible to F if there is a Borel function f with xEy if and only if f(x)Ff(y). Moreover, E is orbit equivalent to F if the respective measure spaces equipped with the extra structure provided by the equivalence relations are almost everywhere isomorphic. We consider product groups acting ergodically and by measure preserving transformations on standard Borel probability spaces. In general terms, the basic parts of the monograph show that if the groups involved have a suitable notion of “boundary" (we make this precise with the definition of near hyperbolic), then one orbit equivalence relation can only be Borel reduced to another if there is some kind of algebraic resemblance between the product groups and coupling of the action. This also has consequence for orbit equivalence. In the case that the original equivalence relations do not have non-trivial almost invariant sets, the techniques lead to relative ergodicity results. An equivalence relation E is said to be relatively ergodic to F if any f with xEy⇒ f(x)Ff(y) has [f(x)]F constant almost everywhere. This underlying collection of lemmas and structural theorems is employed in a number of different ways. One of the most pressing concerns was to give completely self-contained proofs of results which had previously only been obtained using Zimmer's superrigidity theory. We present "elementary proofs" that there are incomparable countable Borel equivalence relations (Adams-Kechris), inclusion does not imply reducibility (Adams), and (n + 1)E is not necessarily reducible to nE (Thomas). In the later parts of the paper we give applications of the theory to specific cases of product groups. In particular, we catalog the actions of products of the free group and obtain additional rigidity theorems and relative ergodicity results in this context. There is a rather long series of appendices, whose primary goal is to give the reader a comprehensive account of the basic techniques. But included here are also some new results. For instance, we show that the Furstenberg-Zimmer lemma on cocycles from amenable groups fails with respect to Baire category, and use this to answer a question of Weiss. We also present a different proof that F_2 has the Haagerup approximation property.