一致超有限性
Uniform hyperfiniteness
摘要：
大约40年前，Connes、Feldman和Weiss证明了对于可测等价关系，顺从性和超有限性的概念是一致的。本文定义了有界顶点度的可测图等价关系的顺从性和超有限性的统一形式，并证明了这两个概念是一致的。粗略地说，一个可测图$\cG$是一致超有限的，如果对任何$\eps>0$，存在$K\geq 1$使得不仅$\cG$，而且它的所有正测度子图都是$(\eps,K)$ -超有限的.我们还表明，这个条件是等价的加权超有限性和强版本的分数超有限性，一个概念最近推出的Lovász。作为推论，我们利用一致超有限性得到了有限生成群正合性的一个刻画。
英文摘要：
Almost forty years ago, Connes, Feldman and Weiss proved that for measurable equivalence relations the notions of amenability and hyperfiniteness coincide. In this paper we define the uniform version of amenability and hyperfiniteness for measurable graphed equivalence relations of bounded vertex degrees and prove that these two notions coincide as well. Roughly speaking, a measured graph $\cG$ is uniformly hyperfinite if for any $\eps>0$ there exists $K\geq 1$ such that not only $\cG$, but all of its subgraphs of positive measure are $(\eps,K)$-hyperfinite. We also show that this condition is equivalent to weighted hyperfiniteness and a strong version of fractional hyperfiniteness, a notion recently introduced by Lovász. As a corollary, we obtain a characterization of exactness of finitely generated groups via uniform hyperfiniteness.
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